A graph on <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063904pt" id="M2" height="8.55521pt" version="1.1" viewBox="-0.0498162 -8.34882 12.9135 8.55521" width="12.9135pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-51" d="M412 140C382 77 369 73 315 73H129L270 222C362 320 402 379 402 466C402 571 322 635 234 635C177 635 130 609 99 576L42 495L64 475C90 514 133 568 201 568C274 568 318 519 318 435C318 349 255 267 193 193C144 135 87 78 32 23V0H405C417 45 427 89 440 131L412 140Z"/></g><g transform="matrix(.013,0,0,-0.013,6.24,0)"><path id="g113-111" d="M495 86L479 114C446 82 419 66 409 66C401 66 401 72 406 97C420 166 436 231 453 297C489 435 454 448 428 448C406 448 384 439 354 422C305 394 222 327 161 247H159L183 345C200 415 194 448 173 448C143 448 82 410 23 351L38 325C64 349 95 371 105 371C111 371 116 365 109 336L25 -4L31 -12C50 -4 77 3 107 9C119 69 132 122 145 168C197 254 321 381 370 381C387 381 393 374 378 305L329 95C309 17 320 -12 345 -12C372 -12 430 19 495 86Z"/></g></svg> vertices can be starter-labelled, if the vertices can be given labels from the nonzero elements of the additive group <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.40071pt" id="M3" height="12.0888pt" version="1.1" viewBox="-0.0498162 -8.68809 29.2384 12.0888" width="29.2384pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g197-37" d="M677 618V662H100V618H469L50 44V0H668V44H252L677 618ZM617 618L200 44H103L523 618H617Z"/></g><g transform="matrix(.0091,0,0,-0.0091,9.451,3.132)"><path id="g50-51" d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z"/></g><g transform="matrix(.0091,0,0,-0.0091,13.883,3.132)"><path id="g50-111" d="M504 89L488 119C453 85 428 69 416 69C408 69 405 75 412 102L462 304C492 438 460 451 436 451S388 441 356 423C311 397 229 336 170 256H168L189 340C208 418 200 451 177 451C144 451 82 413 24 352L40 322C65 345 98 369 108 369C114 369 119 363 112 335L28 -3L36 -12C54 -3 83 4 112 10C125 73 138 122 153 174C205 258 328 382 376 382C395 382 399 369 384 305L330 93C312 25 323 -12 351 -12C378 -12 439 21 504 89Z"/></g><g transform="matrix(.0091,0,0,-0.0091,18.542,3.132)"><path id="g54-36" d="M556 236V289H337V504H275V289H56V236H275V-4H337V236H556Z"/></g><g transform="matrix(.0091,0,0,-0.0091,24.102,3.132)"><path id="g50-50" d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 497V118C207 46 201 38 110 32V0H389Z"/></g></svg> such that each label <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2064009pt" id="M4" height="8.8423pt" version="1.1" viewBox="-0.0498162 -8.6359 3.66193 8.8423" width="3.66193pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-106" d="M244 607C244 633 228 655 200 655C166 655 146 618 146 594C146 564 166 546 191 546C221 546 244 574 244 607ZM222 91L209 114C184 94 148 66 133 66C127 66 124 73 130 96L201 370C213 416 211 448 191 448C162 448 88 407 29 352L42 328C73 354 104 371 114 371C120 371 119 365 115 345L53 92C32 5 45 -12 68 -12C103 -12 186 50 222 91Z"/></g></svg>, either <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2064009pt" id="M5" height="8.8423pt" version="1.1" viewBox="-0.0498162 -8.6359 3.66193 8.8423" width="3.66193pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-106"/></g></svg> or <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063999pt" id="M6" height="11.7978pt" version="1.1" viewBox="-0.0498162 -11.5914 14.1893 11.7978" width="14.1893pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-106"/></g><g transform="matrix(.0091,0,0,-0.0091,3.549,-5.741)"><path id="g54-33" d="M556 236V289H56V236H556Z"/></g><g transform="matrix(.0091,0,0,-0.0091,9.109,-5.741)"><use xlink:href="#g50-50"/></g></svg>, is assigned to exactly two vertices and the two vertices are separated by either <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2064009pt" id="M7" height="8.8423pt" version="1.1" viewBox="-0.0498162 -8.6359 3.66193 8.8423" width="3.66193pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-106"/></g></svg> edges or <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063999pt" id="M8" height="11.7978pt" version="1.1" viewBox="-0.0498162 -11.5914 14.1893 11.7978" width="14.1893pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-106"/></g><g transform="matrix(.0091,0,0,-0.0091,3.549,-5.741)"><use xlink:href="#g54-33"/></g><g transform="matrix(.0091,0,0,-0.0091,9.109,-5.741)"><use xlink:href="#g50-50"/></g></svg> edges, respectively. Mendelsohn and Shalaby have introduced Skolem-labelled graphs and determined the conditions of <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063999pt" id="M9" height="9.49473pt" version="1.1" viewBox="-0.0498162 -9.28833 6.66314 9.49473" width="6.66314pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-108"/></g></svg>-windmills to be Skolem-labelled. In this paper, we introduce starter-labelled graphs and obtain necessary and sufficient conditions for starter and minimum hooked starter labelling of all <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063999pt" id="M10" height="9.49473pt" version="1.1" viewBox="-0.0498162 -9.28833 6.66314 9.49473" width="6.66314pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-108"/></g></svg>-windmills.

Farej Omer

Nabil Shalaby

Starter Labelling of <svg  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063999pt" height="9.49473pt" version="1.1" viewBox="-0.0498162 -9.28833 6.66314 9.49473" width="6.66314pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-108" d="M480 416C480 431 465 448 438 448C388 448 312 383 252 330C217 299 188 273 155 237H153L257 680C262 700 263 712 253 712C240 712 183 684 97 674L92 648L126 647C166 646 172 645 163 606L23 -6L29 -12C51 -5 77 2 107 8C115 62 130 128 142 180C153 193 179 220 204 241C231 170 259 106 288 54C317 0 336 -12 358 -12C381 -12 423 2 477 80L460 100C434 74 408 54 398 54C385 54 374 65 351 107C326 154 282 241 263 299C296 332 351 377 403 377C424 377 436 372 445 368C449 366 456 368 462 375C472 386 480 402 480 416Z"/></g></svg>-Windmill Graphs with Small Defects

We derive a formula for the reliability of a <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063999pt" id="M4" height="9.49473pt" version="1.1" viewBox="-0.0498162 -9.28833 7.34169 9.49473" width="7.34169pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-101"/></g></svg>-dimensional consecutive-<svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063999pt" id="M5" height="9.49473pt" version="1.1" viewBox="-0.0498162 -9.28833 6.66314 9.49473" width="6.66314pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-108"/></g></svg>-out-of-<svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063904pt" id="M6" height="6.1673pt" version="1.1" viewBox="-0.0498162 -5.96091 6.6501 6.1673" width="6.6501pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-111"/></g></svg>:F system, that is, a formula for the probability that an <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.291101pt" id="M7" height="10.2959pt" version="1.1" viewBox="-0.0498162 -7.0048 63.891 10.2959" width="63.891pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-111"/></g><g transform="matrix(.0091,0,0,-0.0091,6.461,3.132)"><path id="g50-50" d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 497V118C207 46 201 38 110 32V0H389Z"/></g><g transform="matrix(.013,0,0,-0.013,14.313,0)"><path id="g117-42" d="M528 54L331 254L528 455L492 493L294 291L96 493L60 455L257 254L60 54L96 16L294 217L492 16L528 54Z"/></g><g transform="matrix(.013,0,0,-0.013,24.849,0)"><path id="g113-46" d="M170 255C170 288 146 313 114 313S58 288 58 255C58 221 82 198 114 198S170 221 170 255Z"/></g><g transform="matrix(.013,0,0,-0.013,29.992,0)"><use xlink:href="#g113-46"/></g><g transform="matrix(.013,0,0,-0.013,35.136,0)"><use xlink:href="#g113-46"/></g><g transform="matrix(.013,0,0,-0.013,41.038,0)"><use xlink:href="#g117-42"/></g><g transform="matrix(.013,0,0,-0.013,51.575,0)"><use xlink:href="#g113-111"/></g><g transform="matrix(.0091,0,0,-0.0091,58.036,3.132)"><path id="g50-101" d="M538 684C542 703 539 710 529 710C512 710 455 682 363 674L361 644H396C444 644 448 642 438 594L405 438C373 450 347 451 336 451C288 451 200 416 147 374C69 312 24 207 24 113C24 29 60 -12 94 -12C126 -12 156 4 196 30C231 53 296 100 346 166H348L328 81C311 10 325 -12 348 -12C380 -12 450 24 512 98L495 125C466 97 431 70 417 70C408 70 408 80 411 98L538 684ZM390 373L361 240C336 196 210 56 145 56C132 56 113 73 113 131C113 216 156 338 220 379C243 394 266 402 303 402C331 402 375 388 390 373Z"/></g></svg> array whose entries are (independently of each other) 0 with probability <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.42943pt" id="M8" height="10.2124pt" version="1.1" viewBox="-0.0498162 -6.78297 7.83752 10.2124" width="7.83752pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-113" d="M570 304C570 398 525 448 414 448C385 448 343 445 312 434L329 511L321 518C297 504 262 482 244 460L233 411C195 397 159 381 128 358L135 332C160 347 189 360 224 373L111 -147C97 -210 84 -218 17 -231L13 -257L254 -247L259 -218L233 -216C183 -212 177 -202 189 -142L218 -1C238 -10 266 -12 283 -12C351 3 429 48 483 105C543 168 570 242 570 304ZM482 289C482 161 380 33 304 33C278 33 248 51 233 69L303 396C326 400 352 403 369 403C428 403 482 380 482 289Z"/></g></svg> and 1 with probability <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.42938pt" id="M9" height="11.7782pt" version="1.1" viewBox="-0.0498162 -8.34882 48.9018 11.7782" width="48.9018pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-114" d="M474 429L457 433C435 440 389 448 367 448C348 448 323 446 309 443C266 434 195 406 148 366C78 307 23 210 23 101C23 35 55 -12 92 -12C118 -12 146 1 196 35C247 70 311 130 346 173H348L281 -148C268 -211 256 -221 208 -229L191 -232L187 -257L433 -245L437 -219L411 -216C357 -210 350 -205 362 -140L427 207C447 315 461 381 474 429ZM387 387C379 337 363 262 355 236C318 180 201 57 142 57C126 57 112 81 112 128C112 205 150 321 220 376C244 395 280 403 312 403C345 403 370 396 387 387Z"/></g><g transform="matrix(.013,0,0,-0.013,10.015,0)"><path id="g117-34" d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"/></g><g transform="matrix(.013,0,0,-0.013,21.278,0)"><path id="g113-50" d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z"/></g><g transform="matrix(.013,0,0,-0.013,30.424,0)"><path id="g117-33" d="M535 230V280H52V230H535Z"/></g><g transform="matrix(.013,0,0,-0.013,40.96,0)"><use xlink:href="#g113-113"/></g></svg> does not include a contiguous <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.291101pt" id="M10" height="10.2959pt" version="1.1" viewBox="-0.0498162 -7.0048 60.5766 10.2959" width="60.5766pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-116" d="M352 391C352 416 319 448 267 448C236 448 173 423 147 400C107 364 96 332 96 304C96 248 143 210 193 181C241 153 258 124 258 100C258 72 232 38 184 38C151 38 107 66 81 108C77 114 64 116 55 111C34 99 23 84 23 65C23 29 81 -12 134 -12C220 -12 325 61 325 141C325 184 297 215 234 256C194 282 161 309 161 346C161 380 188 401 217 401C255 401 279 380 301 353C308 344 313 341 325 347C341 355 352 371 352 391Z"/></g><g transform="matrix(.0091,0,0,-0.0091,4.81,3.132)"><use xlink:href="#g50-50"/></g><g transform="matrix(.013,0,0,-0.013,12.662,0)"><use xlink:href="#g117-42"/></g><g transform="matrix(.013,0,0,-0.013,23.198,0)"><use xlink:href="#g113-46"/></g><g transform="matrix(.013,0,0,-0.013,28.342,0)"><use xlink:href="#g113-46"/></g><g transform="matrix(.013,0,0,-0.013,33.485,0)"><use xlink:href="#g113-46"/></g><g transform="matrix(.013,0,0,-0.013,39.388,0)"><use xlink:href="#g117-42"/></g><g transform="matrix(.013,0,0,-0.013,49.924,0)"><use xlink:href="#g113-116"/></g><g transform="matrix(.0091,0,0,-0.0091,54.734,3.132)"><use xlink:href="#g50-101"/></g></svg> subarray whose every entry is 1.

Simon Cowell

A Formula for the Reliability of a <svg  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063999pt" height="9.49473pt" version="1.1" viewBox="-0.0498162 -9.28833 7.34169 9.49473" width="7.34169pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-101" d="M530 686C535 705 530 712 521 712C504 712 448 684 359 674L358 648H393C437 648 439 646 429 593L400 435C372 447 345 448 332 448C286 448 194 414 144 373C68 311 23 203 23 111C23 26 57 -12 91 -12C120 -12 147 3 188 29C227 54 290 102 341 170H343L322 71C308 6 320 -12 341 -12C373 -12 442 27 501 96L485 120C455 91 422 67 408 67C401 67 401 76 404 91C440 294 479 473 530 686ZM387 375L355 241C326 187 200 53 142 53C126 53 109 73 109 130C109 217 154 337 218 381C240 396 265 404 297 404S372 390 387 375Z"/></g></svg>-Dimensional Consecutive-<svg  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063999pt" height="9.49473pt" version="1.1" viewBox="-0.0498162 -9.28833 6.66314 9.49473" width="6.66314pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-108" d="M480 416C480 431 465 448 438 448C388 448 312 383 252 330C217 299 188 273 155 237H153L257 680C262 700 263 712 253 712C240 712 183 684 97 674L92 648L126 647C166 646 172 645 163 606L23 -6L29 -12C51 -5 77 2 107 8C115 62 130 128 142 180C153 193 179 220 204 241C231 170 259 106 288 54C317 0 336 -12 358 -12C381 -12 423 2 477 80L460 100C434 74 408 54 398 54C385 54 374 65 351 107C326 154 282 241 263 299C296 332 351 377 403 377C424 377 436 372 445 368C449 366 456 368 462 375C472 386 480 402 480 416Z"/></g></svg>-out-of-<svg  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063904pt" height="6.1673pt" version="1.1" viewBox="-0.0498162 -5.96091 6.6501 6.1673" width="6.6501pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-111" d="M495 86L479 114C446 82 419 66 409 66C401 66 401 72 406 97C420 166 436 231 453 297C489 435 454 448 428 448C406 448 384 439 354 422C305 394 222 327 161 247H159L183 345C200 415 194 448 173 448C143 448 82 410 23 351L38 325C64 349 95 371 105 371C111 371 116 365 109 336L25 -4L31 -12C50 -4 77 3 107 9C119 69 132 122 145 168C197 254 321 381 370 381C387 381 393 374 378 305L329 95C309 17 320 -12 345 -12C372 -12 430 19 495 86Z"/></g></svg>:F System

The <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063999pt" id="M1" height="6.20645pt" version="1.1" viewBox="-0.0498162 -6.00005 6.78059 6.20645" width="6.78059pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g50-111" d="M504 89L488 119C453 85 428 69 416 69C408 69 405 75 412 102L462 304C492 438 460 451 436 451S388 441 356 423C311 397 229 336 170 256H168L189 340C208 418 200 451 177 451C144 451 82 413 24 352L40 322C65 345 98 369 108 369C114 369 119 363 112 335L28 -3L36 -12C54 -3 83 4 112 10C125 73 138 122 153 174C205 258 328 382 376 382C395 382 399 369 384 305L330 93C312 25 323 -12 351 -12C378 -12 439 21 504 89Z"/></g></svg>-dimensional hypercube <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.16057pt" id="M2" height="11.9661pt" version="1.1" viewBox="-0.0498162 -8.80553 14.7087 11.9661" width="14.7087pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g50-82" d="M750 355C750 552 603 667 428 667C199 667 24 497 24 286C24 118 146 -4 316 -16L336 -29C480 -122 544 -145 582 -154C606 -159 666 -170 714 -172L722 -146C652 -117 562 -75 472 -15L454 -3C623 38 750 176 750 355ZM646 353C646 210 556 50 414 25L385 41L304 27C193 50 128 152 128 282C128 474 254 626 418 626C571 626 646 519 646 353Z"/></g><g transform="matrix(.0091,0,0,-0.0091,9.399,3.001)"><use xlink:href="#g50-111"/></g></svg> is  <i>bipancyclic</i>; that is, it contains a cycle of every even length from 4 to <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.04979992pt" id="M3" height="10.1367pt" version="1.1" viewBox="-0.0498162 -10.0869 11.6292 10.1367" width="11.6292pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g50-51" d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z"/></g><g transform="matrix(.0091,0,0,-0.0091,6.331,-5.872)"><use xlink:href="#g50-111"/></g></svg>. In this paper, we prove that <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.16067pt" id="M4" height="12.449pt" version="1.1" viewBox="-0.0498162 -9.28833 56.8211 12.449" width="56.8211pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g50-82"/></g><g transform="matrix(.0091,0,0,-0.0091,9.399,3.001)"><use xlink:href="#g50-111"/></g><g transform="matrix(.013,0,0,-0.013,18.984,0)"><path id="g50-41" d="M309 -142C211 -61 151 99 151 271S210 601 309 683L288 710C141 611 75 451 75 272V271C75 90 141 -72 288 -169L309 -142Z"/></g><g transform="matrix(.013,0,0,-0.013,23.638,0)"><use xlink:href="#g50-111"/></g><g transform="matrix(.013,0,0,-0.013,33.97,0)"><path id="g54-94" d="M552 293V344L59 566V507L489 319L59 131V72L552 293ZM552 -41V12H59V-41H552Z"/></g><g transform="matrix(.013,0,0,-0.013,45.588,0)"><path id="g50-52" d="M290 377C321 398 342 415 358 430C378 450 389 473 389 502C389 578 329 635 238 635H237C184 635 137 610 109 578L64 515L88 493C112 529 154 573 208 573S303 542 303 482C303 409 233 370 141 341L149 308C165 313 190 319 215 319C272 319 341 283 341 193C342 98 292 43 222 43C163 43 122 72 96 94C88 101 79 100 70 94C61 87 47 73 46 60C44 47 48 37 62 23C76 10 118 -12 165 -12C238 -12 430 62 430 223C430 297 379 359 290 375V377Z"/></g><g transform="matrix(.013,0,0,-0.013,51.919,0)"><path id="g50-42" d="M283 271C283 451 220 610 70 710L48 683C146 603 207 443 207 271S149 -59 48 -142L70 -169C220 -67 283 90 283 271Z"/></g></svg> contains a 3-regular, 3-connected, bipancyclic subgraph with <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063999pt" id="M5" height="9.49473pt" version="1.1" viewBox="-0.0498162 -9.28833 3.63583 9.49473" width="3.63583pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g50-109" d="M238 675C244 702 241 710 231 710C220 710 155 682 74 673L70 643H101C147 643 151 635 142 594L39 111C18 11 36 -12 55 -12C89 -12 170 38 223 103L207 129C176 98 130 68 118 68C113 68 108 72 114 102L238 675Z"/></g></svg> vertices for every even <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063999pt" id="M6" height="9.49473pt" version="1.1" viewBox="-0.0498162 -9.28833 3.63583 9.49473" width="3.63583pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g50-109"/></g></svg> from 8 to <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.04979992pt" id="M7" height="10.1367pt" version="1.1" viewBox="-0.0498162 -10.0869 11.6292 10.1367" width="11.6292pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g50-51"/></g><g transform="matrix(.0091,0,0,-0.0091,6.331,-5.872)"><use xlink:href="#g50-111"/></g></svg> except 10.

Y. M. Borse

S. R. Shaikh

On 3-Regular Bipancyclic Subgraphs of Hypercubes

Aras Erzurumluo&#x11F;lu

C. A. Rodger

On Evenly-Equitable, Balanced Edge-Colorings and Related Notions

Much research has involved the consideration of graphs which have subgraphs of a particular kind, such as cliques.  Known classes of graphs which are eigen-bi-balanced, that is, they have a pair <i>a</i>, <i>b</i> of  nonzero distinct eigenvalues, whose sum and product are integral, have been investigated.  In this paper we will define a new class of graphs, called <i>q-cliqued</i> graphs, on <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.571899pt" id="M2" height="15.65pt" version="1.1" viewBox="-0.0498162 -12.0781 32.4769 15.65" width="32.4769pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-114"/></g><g transform="matrix(.0095,0,0,-0.0095,6.652,-5.984)"><path id="g50-51" d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z"/></g><g transform="matrix(.0135,0,0,-0.0135,14.814,0)"><path id="g117-36" d="M535 230V280H323V490H265V280H52V230H265V-3H323V230H535Z"/></g><g transform="matrix(.0135,0,0,-0.0135,25.795,0)"><path id="g113-50" d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z"/></g></svg> vertices, which contain <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.57194pt" id="M3" height="9.7821pt" version="1.1" viewBox="-0.0498162 -6.21016 6.77673 9.7821" width="6.77673pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-114"/></g></svg> cliques each of order <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.57194pt" id="M4" height="9.7821pt" version="1.1" viewBox="-0.0498162 -6.21016 6.77673 9.7821" width="6.77673pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-114"/></g></svg> connected to a central vertex, and then prove that these <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.57194pt" id="M5" height="9.7821pt" version="1.1" viewBox="-0.0498162 -6.21016 6.77673 9.7821" width="6.77673pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-114"/></g></svg>-cliqued graphs are eigen-bi-balanced with respect to a conjugate pair whose sum is <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.3625803pt" id="M6" height="9.06134pt" version="1.1" viewBox="-0.0498162 -8.69876 14.6097 9.06134" width="14.6097pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g117-33" d="M535 230V280H52V230H535Z"/></g><g transform="matrix(.0135,0,0,-0.0135,7.953,0)"><use xlink:href="#g113-50"/></g></svg> and product <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.57194pt" id="M7" height="12.2707pt" version="1.1" viewBox="-0.0498162 -8.69876 27.3429 12.2707" width="27.3429pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-50"/></g><g transform="matrix(.0135,0,0,-0.0135,9.531,0)"><use xlink:href="#g117-33"/></g><g transform="matrix(.0135,0,0,-0.0135,20.512,0)"><use xlink:href="#g113-114"/></g></svg>. These graphs can be regarded as design graphs, and we use a specific example in an entomological experiment.

Paul August Winter

Carol Lynne Jessop

Costas Zachariades

The Class of <svg  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.57194pt" height="9.7821pt" version="1.1" viewBox="-0.0498162 -6.21016 6.77673 9.7821" width="6.77673pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-114" d="M474 429L457 433C435 440 389 448 367 448C348 448 323 446 309 443C266 434 195 406 148 366C78 307 23 210 23 101C23 35 55 -12 92 -12C118 -12 146 1 196 35C247 70 311 130 346 173H348L281 -148C268 -211 256 -221 208 -229L191 -232L187 -257L433 -245L437 -219L411 -216C357 -210 350 -205 362 -140L427 207C447 315 461 381 474 429ZM387 387C379 337 363 262 355 236C318 180 201 57 142 57C126 57 112 81 112 128C112 205 150 321 220 376C244 395 280 403 312 403C345 403 370 396 387 387Z" /><glyph.data ascent="3473" descent="-2876" horiz-adv-x="491" vert-adv-y="491" /></g></svg>-Cliqued Graphs: Eigen-Bi-Balanced Characteristic, Designs, and an Entomological Experiment

This paper considers the varietal hypercube network <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.427751pt" id="M1" height="12.4257pt" version="1.1" viewBox="-0.0498162 -8.99795 24.1838 12.4257" width="24.1838pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-87" d="M697 650H468L461 623L492 619C539 613 547 605 518 546C481 471 367 264 278 116H276C239 278 197 500 186 567C180 604 185 613 226 619L252 623L260 650H24L17 623C78 617 92 613 108 533L216 -11H247C365 200 515 462 560 529C616 612 624 615 689 623L697 650Z"/></g><g transform="matrix(.0135,0,0,-0.0135,8.933,0)"><path id="g113-82" d="M699 368C699 549 574 666 407 666C186 666 23 488 23 277C23 113 129 -3 288 -13L307 -26C431 -111 501 -139 533 -147C559 -154 613 -163 658 -164L666 -141C597 -111 507 -66 430 -11L416 -1C580 42 699 190 699 368ZM601 371C601 227 518 54 381 22L354 40L278 24C175 47 120 145 120 269C120 451 235 631 398 631C540 631 601 521 601 371Z"/></g><g transform="matrix(.0095,0,0,-0.0095,18.647,3.264)"><path id="g50-111" d="M504 89L488 119C453 85 428 69 416 69C408 69 405 75 412 102L462 304C492 438 460 451 436 451S388 441 356 423C311 397 229 336 170 256H168L189 340C208 418 200 451 177 451C144 451 82 413 24 352L40 322C65 345 98 369 108 369C114 369 119 363 112 335L28 -3L36 -12C54 -3 83 4 112 10C125 73 138 122 153 174C205 258 328 382 376 382C395 382 399 369 384 305L330 93C312 25 323 -12 351 -12C378 -12 439 21 504 89Z"/></g></svg> with mixed faults and shows that <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.427751pt" id="M2" height="12.4257pt" version="1.1" viewBox="-0.0498162 -8.99795 24.1838 12.4257" width="24.1838pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-87"/></g><g transform="matrix(.0135,0,0,-0.0135,8.933,0)"><use xlink:href="#g113-82"/></g><g transform="matrix(.0095,0,0,-0.0095,18.647,3.264)"><use xlink:href="#g50-111"/></g></svg> contains a fault-free Hamilton cycle provided faults do not exceed <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.3625803pt" id="M3" height="9.06134pt" version="1.1" viewBox="-0.0498162 -8.69876 27.4925 9.06134" width="27.4925pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-111" d="M495 86L479 114C446 82 419 66 409 66C401 66 401 72 406 97C420 166 436 231 453 297C489 435 454 448 428 448C406 448 384 439 354 422C305 394 222 327 161 247H159L183 345C200 415 194 448 173 448C143 448 82 410 23 351L38 325C64 349 95 371 105 371C111 371 116 365 109 336L25 -4L31 -12C50 -4 77 3 107 9C119 69 132 122 145 168C197 254 321 381 370 381C387 381 393 374 378 305L329 95C309 17 320 -12 345 -12C372 -12 430 19 495 86Z"/></g><g transform="matrix(.0135,0,0,-0.0135,9.829,0)"><path id="g117-33" d="M535 230V280H52V230H535Z"/></g><g transform="matrix(.0135,0,0,-0.0135,20.81,0)"><path id="g113-51" d="M412 140C382 77 369 73 315 73H129L270 222C362 320 402 379 402 466C402 571 322 635 234 635C177 635 130 609 99 576L42 495L64 475C90 514 133 568 201 568C274 568 318 519 318 435C318 349 255 267 193 193C144 135 87 78 32 23V0H405C417 45 427 89 440 131L412 140Z"/></g></svg> for <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-1.16493pt" id="M4" height="9.86369pt" version="1.1" viewBox="-0.0498162 -8.69876 29.0065 9.86369" width="29.0065pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-111"/></g><g transform="matrix(.0135,0,0,-0.0135,10.586,0)"><path id="g117-96" d="M531 317V367L57 582V526L473 342L57 159V103L531 317ZM531 143V199L57 -15V-71L531 143Z"/></g><g transform="matrix(.0135,0,0,-0.0135,22.324,0)"><use xlink:href="#g113-51"/></g></svg> and contains a fault-free Hamilton path between any pair of vertices provided faults do not exceed <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.3625803pt" id="M5" height="9.06134pt" version="1.1" viewBox="-0.0498162 -8.69876 27.4925 9.06134" width="27.4925pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-111"/></g><g transform="matrix(.0135,0,0,-0.0135,9.829,0)"><use xlink:href="#g117-33"/></g><g transform="matrix(.0135,0,0,-0.0135,20.81,0)"><path id="g113-52" d="M285 378C315 398 338 416 353 432C373 451 384 474 384 503C384 579 325 635 236 635H235C182 635 136 610 108 579L65 516L85 496C110 533 150 575 205 575C258 575 300 543 300 481C300 407 232 369 141 339L147 310C163 315 188 321 211 321C268 321 338 284 338 192C338 94 288 40 217 40C160 40 119 68 93 91C85 98 77 97 69 91C60 84 47 71 46 58C44 46 48 35 62 22C75 10 116 -12 162 -12C234 -12 424 62 424 224C424 297 373 359 285 376V378Z"/></g></svg> for <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-1.16493pt" id="M6" height="9.86369pt" version="1.1" viewBox="-0.0498162 -8.69876 29.0065 9.86369" width="29.0065pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-111"/></g><g transform="matrix(.0135,0,0,-0.0135,10.586,0)"><use xlink:href="#g117-96"/></g><g transform="matrix(.0135,0,0,-0.0135,22.324,0)"><use xlink:href="#g113-52"/></g></svg>. The proof is based on an inductive construction.

Jian-Guang Zhou

Jun-Ming Xu

Hamilton Paths and Cycles in Varietal Hypercube Networks with Mixed Faults

A set <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.6617708pt" id="M1" height="9.71411pt" version="1.1" viewBox="-0.0498162 -9.05234 31.9574 9.71411" width="31.9574pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-84" d="M449 634C442 637 425 643 405 650C376 660 341 666 307 666C181 666 98 590 98 485C98 400 170 343 215 310L246 288C307 243 343 204 343 147C343 67 291 18 219 18C104 18 61 124 51 202L23 199C28 124 27 71 27 47C47 22 122 -16 204 -16C324 -16 428 60 428 174C428 256 379 309 307 360L276 382C223 419 179 455 179 516C179 576 221 632 293 632C379 632 410 564 418 487L448 490C446 536 446 592 449 634Z"/></g><g transform="matrix(.0135,0,0,-0.0135,10.18,0)"><path id="g117-77" d="M512 1V51H381C244 51 133 142 133 255S244 459 381 459H512V509H382C213 509 75 395 75 255S213 1 382 1H512Z"/></g><g transform="matrix(.0136,0,0,-0.0136,21.918,0)"><path id="g197-37" d="M677 618V662H100V618H469L50 44V0H668V44H252L677 618ZM617 618L200 44H103L523 618H617Z"/></g></svg> is midpoint-free if no ordered triple <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.3616pt" id="M2" height="14.4397pt" version="1.1" viewBox="-0.0498162 -12.0781 65.3364 14.4397" width="65.3364pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-41" d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"/></g><g transform="matrix(.0135,0,0,-0.0135,4.688,0)"><path id="g113-98" d="M483 97L471 123C436 91 401 65 392 65C388 65 384 74 390 106C414 239 444 378 457 429L455 433C444 433 429 436 416 439C392 444 368 448 344 448C281 448 204 415 152 376C71 315 23 205 23 103C23 21 57 -12 85 -12C114 -12 149 6 185 34C231 70 285 119 329 183H331L309 81C292 0 308 -12 326 -12C350 -12 421 24 483 97ZM374 387C370 363 356 291 345 261C315 193 181 50 139 50C124 50 110 71 110 118C110 224 153 331 218 379C238 394 271 402 301 402C329 402 359 394 374 387Z"/></g><g transform="matrix(.0135,0,0,-0.0135,11.545,0)"><path id="g113-45" d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z"/></g><g transform="matrix(.0135,0,0,-0.0135,16.905,0)"><path id="g113-99" d="M452 333C452 398 425 448 375 448C329 448 235 404 140 292H138L229 683C233 701 234 712 225 712C208 712 149 686 66 677L64 652H97C135 652 140 651 129 598L38 167C23 96 29 57 44 33C60 7 98 -12 132 -12C169 -12 232 3 290 41C383 102 452 223 452 333ZM365 316C365 199 308 87 239 49C221 39 200 35 183 35C137 35 99 70 112 163C116 191 123 215 129 233C190 316 290 392 330 392C348 392 365 372 365 316Z"/></g><g transform="matrix(.0135,0,0,-0.0135,23.264,0)"><use xlink:href="#g113-45"/></g><g transform="matrix(.0135,0,0,-0.0135,28.624,0)"><path id="g113-100" d="M383 397C383 424 344 448 299 448C244 448 172 409 132 375C66 319 23 227 23 146C23 42 74 -12 146 -12C208 -12 298 30 359 103L343 124C315 95 248 48 192 48C145 48 111 85 111 163C111 228 129 294 151 330C171 363 201 401 241 401C275 401 302 384 325 356C332 347 339 344 348 348C373 360 383 381 383 397Z"/></g><g transform="matrix(.0135,0,0,-0.0135,34.129,0)"><path id="g113-42" d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"/></g><g transform="matrix(.0135,0,0,-0.0135,42.601,0)"><path id="g117-173" d="M448 1V51H364C248 51 153 129 140 230H448V280H140C153 381 248 459 364 459H448V509H365C208 509 80 395 80 255S208 1 365 1H448Z"/></g><g transform="matrix(.0135,0,0,-0.0135,53.54,0)"><use xlink:href="#g113-84"/></g><g transform="matrix(.0095,0,0,-0.0095,59.935,-5.984)"><path id="g50-52" d="M290 377C321 398 342 415 358 430C378 450 389 473 389 502C389 578 329 635 238 635H237C184 635 137 610 109 578L64 515L88 493C112 529 154 573 208 573S303 542 303 482C303 409 233 370 141 341L149 308C165 313 190 319 215 319C272 319 341 283 341 193C342 98 292 43 222 43C163 43 122 72 96 94C88 101 79 100 70 94C61 87 47 73 46 60C44 47 48 37 62 23C76 10 118 -12 165 -12C238 -12 430 62 430 223C430 297 379 359 290 375V377Z"/></g></svg> satisfies <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.3625994pt" id="M3" height="10.0405pt" version="1.1" viewBox="-0.0498162 -9.6779 54.9986 10.0405" width="54.9986pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-98"/></g><g transform="matrix(.0135,0,0,-0.0135,9.886,0)"><path id="g117-36" d="M535 230V280H323V490H265V280H52V230H265V-3H323V230H535Z"/></g><g transform="matrix(.0135,0,0,-0.0135,20.866,0)"><use xlink:href="#g113-100"/></g><g transform="matrix(.0135,0,0,-0.0135,30.156,0)"><path id="g117-34" d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"/></g><g transform="matrix(.0135,0,0,-0.0135,41.894,0)"><path id="g113-51" d="M412 140C382 77 369 73 315 73H129L270 222C362 320 402 379 402 466C402 571 322 635 234 635C177 635 130 609 99 576L42 495L64 475C90 514 133 568 201 568C274 568 318 519 318 435C318 349 255 267 193 193C144 135 87 78 32 23V0H405C417 45 427 89 440 131L412 140Z"/></g><g transform="matrix(.0135,0,0,-0.0135,48.397,0)"><use xlink:href="#g113-99"/></g></svg> and <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.6617994pt" id="M4" height="10.3397pt" version="1.1" viewBox="-0.0498162 -9.6779 49.9851 10.3397" width="49.9851pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-98"/></g><g transform="matrix(.0135,0,0,-0.0135,10.643,0)"><path id="g117-91" d="M512 -3V55L134 254V256L512 456V514L75 281V230L512 -3Z"/></g><g transform="matrix(.0135,0,0,-0.0135,22.38,0)"><use xlink:href="#g113-99"/></g><g transform="matrix(.0135,0,0,-0.0135,32.524,0)"><use xlink:href="#g117-91"/></g><g transform="matrix(.0135,0,0,-0.0135,44.262,0)"><use xlink:href="#g113-100"/></g></svg>. Midpoint-free subsets of <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.04979992pt" id="M5" height="11.2997pt" version="1.1" viewBox="-0.0498162 -11.2499 16.3005 11.2997" width="16.3005pt"><g transform="matrix(.0136,0,0,-0.0136,0,0)"><use xlink:href="#g197-37"/></g><g transform="matrix(.0095,0,0,-0.0095,9.85,-5.984)"><path id="g54-36" d="M556 236V289H337V504H275V289H56V236H275V-4H337V236H556Z"/></g></svg> and <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.0498209pt" id="M6" height="9.10216pt" version="1.1" viewBox="-0.0498162 -9.05234 9.9861 9.10216" width="9.9861pt"><g transform="matrix(.0136,0,0,-0.0136,0,0)"><use xlink:href="#g197-37"/></g></svg> are studied, with emphasis on those sets characterized by restrictions on the base <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2129903pt" id="M7" height="6.42315pt" version="1.1" viewBox="-0.0498162 -6.21016 10.8292 6.42315" width="10.8292pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-110" d="M766 88L752 113C719 83 690 64 681 64C674 64 672 74 679 103L724 292C758 436 724 448 701 448C680 448 666 442 639 429C594 407 514 350 441 252H439L447 289C476 423 450 448 419 448C398 448 379 441 355 427C307 400 234 344 162 249H160L180 324C203 409 197 448 170 448C144 448 82 413 23 349L35 321C57 343 96 374 108 374C115 374 117 371 111 341C87 227 57 112 24 -6L32 -12C53 -4 81 4 108 6C119 68 134 128 149 171C177 229 309 383 364 383C387 383 388 355 373 282C354 190 330 92 303 -6L309 -12C332 -4 356 3 386 6C396 63 411 122 424 171C458 236 590 383 642 383C658 383 664 369 652 315L603 91C587 20 593 -12 619 -12C642 -12 708 23 766 88Z"/></g></svg> digits of their elements when <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-1.16493pt" id="M8" height="9.86369pt" version="1.1" viewBox="-0.0498162 -8.69876 61.517 9.86369" width="61.517pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-52" d="M285 378C315 398 338 416 353 432C373 451 384 474 384 503C384 579 325 635 236 635H235C182 635 136 610 108 579L65 516L85 496C110 533 150 575 205 575C258 575 300 543 300 481C300 407 232 369 141 339L147 310C163 315 188 321 211 321C268 321 338 284 338 192C338 94 288 40 217 40C160 40 119 68 93 91C85 98 77 97 69 91C60 84 47 71 46 58C44 46 48 35 62 22C75 10 116 -12 162 -12C234 -12 424 62 424 224C424 297 373 359 285 376V378Z"/></g><g transform="matrix(.0135,0,0,-0.0135,10.288,0)"><path id="g117-93" d="M531 71V127L115 310L531 494V550L57 335V285L531 71ZM531 -40V10H57V-40H531Z"/></g><g transform="matrix(.0135,0,0,-0.0135,22.026,0)"><use xlink:href="#g113-110"/></g><g transform="matrix(.0135,0,0,-0.0135,36.501,0)"><use xlink:href="#g117-93"/></g><g transform="matrix(.0135,0,0,-0.0135,48.239,0)"><path id="g113-50" d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z"/></g><g transform="matrix(.0135,0,0,-0.0135,54.742,0)"><path id="g113-53" d="M456 178V225H360V632H320C217 496 115 347 20 206V178H280V106C280 40 276 34 189 27V0H445V27C364 34 360 39 360 106V178H456ZM280 225H82C149 335 214 431 278 520H280V225Z"/></g></svg>, and with particular attention to maximal midpoint-free subsets with <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.3616pt" id="M9" height="12.0395pt" version="1.1" viewBox="-0.0498162 -9.6779 117.155 12.0395" width="117.155pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-110"/></g><g transform="matrix(.0135,0,0,-0.0135,14.475,0)"><use xlink:href="#g117-173"/></g><g transform="matrix(.0135,0,0,-0.0135,25.413,0)"><path id="g113-124" d="M293 -169V-141C218 -131 209 -90 209 -44C209 19 222 85 222 152C222 207 198 255 139 269V273C199 286 222 334 222 388C222 454 209 523 209 588C209 632 218 671 293 681V709C234 709 148 695 148 577C147 513 155 438 155 372C155 337 152 291 64 285V256C152 250 155 204 155 169C156 105 148 31 148 -41C148 -157 234 -169 293 -169Z"/></g><g transform="matrix(.0135,0,0,-0.0135,30.114,0)"><use xlink:href="#g113-52"/></g><g transform="matrix(.0135,0,0,-0.0135,36.618,0)"><use xlink:href="#g113-45"/></g><g transform="matrix(.0135,0,0,-0.0135,41.978,0)"><use xlink:href="#g113-53"/></g><g transform="matrix(.0135,0,0,-0.0135,48.481,0)"><use xlink:href="#g113-45"/></g><g transform="matrix(.0135,0,0,-0.0135,53.841,0)"><path id="g113-56" d="M447 623H65C61 580 56 530 47 475H76C100 541 106 550 172 550H388C308 376 196 170 91 -1L98 -12L172 -2C268 204 360 408 455 611L447 623Z"/></g><g transform="matrix(.0135,0,0,-0.0135,60.347,0)"><use xlink:href="#g113-45"/></g><g transform="matrix(.0135,0,0,-0.0135,65.707,0)"><path id="g113-58" d="M244 635C114 635 38 519 38 422C38 317 111 240 217 240C236 240 255 244 277 256L345 292C311 140 203 39 59 15L64 -15C89 -15 150 -5 204 17C339 72 440 202 440 386C440 521 368 635 244 635ZM228 602C326 602 352 479 352 390C352 370 351 347 348 324C327 308 293 296 258 296C174 296 124 369 124 458C124 517 152 602 228 602Z"/></g><g transform="matrix(.0135,0,0,-0.0135,72.211,0)"><use xlink:href="#g113-45"/></g><g transform="matrix(.0135,0,0,-0.0135,77.571,0)"><path id="g113-47" d="M113 -12C146 -12 170 11 170 46C170 78 146 103 114 103S58 78 58 46C58 11 82 -12 113 -12Z"/></g><g transform="matrix(.0135,0,0,-0.0135,82.931,0)"><use xlink:href="#g113-47"/></g><g transform="matrix(.0135,0,0,-0.0135,88.291,0)"><use xlink:href="#g113-47"/></g><g transform="matrix(.0135,0,0,-0.0135,93.686,0)"><use xlink:href="#g113-45"/></g><g transform="matrix(.0135,0,0,-0.0135,99.046,0)"><use xlink:href="#g113-50"/></g><g transform="matrix(.0135,0,0,-0.0135,105.549,0)"><use xlink:href="#g113-52"/></g><g transform="matrix(.0135,0,0,-0.0135,112.052,0)"><path id="g113-126" d="M283 255V284C195 290 192 337 192 375C192 436 200 508 200 580C200 696 116 709 54 709V681C127 671 139 634 139 588C138 524 125 452 125 387C125 333 149 286 209 272V268C148 253 125 206 125 151C126 87 139 16 139 -47C139 -92 127 -131 54 -141V-169C115 -169 200 -152 200 -41C200 32 192 104 192 164C192 202 195 249 283 255Z"/></g></svg>.

Roger B. Eggleton

Maximal Midpoint-Free Subsets of Integers

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International Journal of Combinatorics