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ABSTRACT A proper conflict‐free coloring of a graph is a coloring of the vertices such that any two adjacent vertices receive different colors, and for every non‐isolated vertex , some color appears exactly once on the neighborhood of . Caro, Petruševski and Škrekovski conjectured that every connected graph with maximum degree has a proper conflict‐free coloring with at most colors. This conjecture holds for and remains open for . In this article we prove that this conjecture holds asymptotically; namely, every graph with maximum degree has a proper conflict‐free coloring with colors.
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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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