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ABSTRACT If is a list assignment of colors to each vertex of an ‐vertex graph , then anequitable‐coloringof is a proper coloring of vertices of from their lists such that no color is used more than times. A graph isequitably‐choosableif it has an equitable ‐coloring for every ‐list assignment . In 2003, Kostochka, Pelsmajer, and West (KPW) conjectured that an analog of the famous Hajnal–Szemerédi Theorem on equitable coloring holds for equitable list coloring, namely, that for each positive integer every graph with maximum degree at most is equitably ‐choosable. The main result of this paper is that for each and each planar graph , a stronger statement holds: if the maximum degree of is at most , then is equitably ‐choosable. In fact, we prove the result for a broader class of graphs—the class of the graphs in which each bipartite subgraph with has at most edges. Together with some known results, this implies that the KPW Conjecture holds for all graphs in , in particular, for all planar graphs. We also introduce the new stronger notion ofstrongly equitable(SE, for short) list coloring and prove all bounds for this parameter. An advantage of this is that if a graph is SE ‐choosable, then it is both equitably ‐choosable and equitably ‐colorable, while neither of being equitably ‐choosable and equitably ‐colorable implies the other.
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This content will become publicly available on November 1, 2026
Total Coloring Graphs With Large Maximum Degree
ABSTRACT We prove that for any graph , the total chromatic number of is at most . This saves one color in comparison with the result of Hind from 1992. In particular, our result says that if , then has a total coloring using at most colors. When is regular and has a sufficient number of vertices, we can actually save an additional two colors. Specifically, we prove that for any , there exists such that: if is an ‐regular graph on vertices with , then . This confirms the Total Coloring Conjecture for such graphs .
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- Award ID(s):
- 2345869
- PAR ID:
- 10648716
- Publisher / Repository:
- WILEY
- Date Published:
- Journal Name:
- Journal of Graph Theory
- Volume:
- 110
- Issue:
- 3
- ISSN:
- 0364-9024
- Page Range / eLocation ID:
- 249 to 262
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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