graph vertic can starterlabel vertic can given label from nonzero element addit group each label either assign exact two vertic two vertic separ either edg edg respect mendelsohn shalabi have introduc skolemlabel graph determin condit windmil skolemlabel paper introduc starterlabel graph obtain necessari suffici condit starter minimum hook starter label all windmil

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:-0.2723999pt" id="M1" height="12.533pt" version="1.1" viewBox="-0.0657574 -12.2606 8.79534 12.533" width="8.79534pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><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

deriv formula reliabl dimension consecut outof system formula probabl array whose entri independ each other 0 probabl 1 probabl doe includ contigu subarray whose everi entri 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:-0.2723999pt" id="M1" height="12.533pt" version="1.1" viewBox="-0.0657574 -12.2606 9.69102 12.533" width="9.69102pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><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:-0.2723999pt" id="M2" height="12.533pt" version="1.1" viewBox="-0.0657574 -12.2606 8.79534 12.533" width="8.79534pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><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:-0.2724395pt" id="M3" height="8.14084pt" version="1.1" viewBox="-0.0657574 -7.8684 8.77813 8.14084" width="8.77813pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><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 dimension hypercub bipancycl contain cycl everi even length from 4 paper prove contain 3regular 3connect bipancycl subgraph vertic everi even from 8 except 10

Y. M. Borse

S. R. Shaikh

On 3-Regular Bipancyclic Subgraphs of Hypercubes

graph said even all vertic have even degre given edgecolor graph each color let denot span subgraph which edgeset contain precis edg color edgecolor said edgecolor each color even graph edgecolor said evenlyequit each color even graph each vertex ani pair color for ani pair vertic let number edg between allow where denot loop incid edgecolor said balanc all pair color all pair vertic possibl hilton prove each even graph has evenlyequit edgecolor each paper extend result find character graph have evenlyequit balanc edgecolor each correspond find character even graph have evenlyequit balanc 2edgecolor then give instanc how evenlyequit balanc edgecolor can use determin certain fair properti factor some regular graph satisfi final indic how differ fair notion edgecolor interact each other

Aras Erzurumluoğlu

C. A. Rodger

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

much research has involv consider graph which have subgraph particular kind cliqu known class graph which eigenbibalanc have pair nonzero distinct eigenvalu whose sum product integr have been investig paper defin new class graph call qcliqu graph vertic which contain cliqu each order connect central vertex prove cliqu graph eigenbibalanc respect conjug pair whose sum product these graph can regard design graph use specif exampl entomolog experi

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:-4.714991pt" id="M1" height="12.9124pt" version="1.1" viewBox="-0.0657574 -8.19741 8.94528 12.9124" width="8.94528pt"><g transform="matrix(.018,0,0,-0.018,0,0)"><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></svg>-Cliqued Graphs: Eigen-Bi-Balanced Characteristic, Designs, and an Entomological Experiment

this paper consid variet hypercub network mix fault show contain faultfre hamilton cycl provid fault exceed contain faultfre hamilton path between ani pair vertic provid fault exceed the proof base induct construct

Jian-Guang Zhou

Jun-Ming Xu

Hamilton Paths and Cycles in Varietal Hypercube Networks with Mixed Faults

set midpointfre order tripl satisfi midpointfre subset studi emphasi those set character restrict base digit element when particular attent maxim midpointfre subset

Roger B. Eggleton

Maximal Midpoint-Free Subsets of Integers

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International Journal of Combinatorics