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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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On the number of edges of separated multigraphs
Abstract We prove that the number of edges of a multigraph with vertices is at most , provided that any two edges cross at most once, parallel edges are noncrossing, and the lens enclosed by every pair of parallel edges in contains at least one vertex. As a consequence, we prove the following extension of the Crossing Lemma of Ajtai, Chvátal, Newborn, Szemerédi, and Leighton, if has edges, in any drawing of with the above property, the number of crossings is . This answers a question of Kaufmann et al. and is tight up to the logarithmic factor.
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- Award ID(s):
- 1800746
- PAR ID:
- 10461993
- Publisher / Repository:
- Wiley Blackwell (John Wiley & Sons)
- Date Published:
- Journal Name:
- Journal of Graph Theory
- ISSN:
- 0364-9024
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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