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

<table class="figure-group"><tr class="fig-image" id="a"><td><object data="https://static.hindawi.com/articles/ijcom/volume-2015/201427/figures/201427.fig.001a.svgz" name="201427.fig.001a" type="image/svg+xml"></object></td></tr><tr class="fig-caption"><td><b>(a) </b><svg height="12.2868pt" id="M607" style="vertical-align:-3.2637pt" version="1.1" viewbox="0 -9.0231 12.8884 12.2868" width="12.8884pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path d="M673 297H417L411 270C517 261 522 258 506 183L487 92C476 38 428 19 360 19C207 19 123 132 123 287C123 472 243 631 441 631C557 631 617 583 617 469L647 472C650 545 655 604 658 632C625 643 554 666 467 666C201 666 23 502 23 278C23 90 156 -16 339 -16C410 -16 501 6 569 24C566 43 567 70 573 101L589 183C604 259 607 262 667 271L673 297Z" id="g113-72"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="684" vert-adv-y="684"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,9.199,3.264)"><path d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 497V118C207 46 201 38 110 32V0H389Z" id="g50-50"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="487" vert-adv-y="487"></glyph.data></g></svg>: A vertex-minimum example</td></tr><tr class="fig-image" id="b"><td><object data="https://static.hindawi.com/articles/ijcom/volume-2015/201427/figures/201427.fig.001b.svgz" name="201427.fig.001b" type="image/svg+xml"></object></td></tr><tr class="fig-caption"><td><b>(b) </b><svg height="12.2868pt" id="M608" style="vertical-align:-3.2637pt" version="1.1" viewbox="0 -9.0231 13.4289 12.2868" width="13.4289pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path d="M673 297H417L411 270C517 261 522 258 506 183L487 92C476 38 428 19 360 19C207 19 123 132 123 287C123 472 243 631 441 631C557 631 617 583 617 469L647 472C650 545 655 604 658 632C625 643 554 666 467 666C201 666 23 502 23 278C23 90 156 -16 339 -16C410 -16 501 6 569 24C566 43 567 70 573 101L589 183C604 259 607 262 667 271L673 297Z" id="g113-72"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="684" vert-adv-y="684"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,9.199,3.264)"><path 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" id="g50-51"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="487" vert-adv-y="487"></glyph.data></g></svg>: An edge-minimum example</td></tr></table>

<div>Examples of graphs that are not of color-type 1.</div>

International Journal of Combinatorics

fig1

Figure 1

Figure 1: On Evenly-Equitable, Balanced Edge-Colorings and Related Notions 