Chromatic number and complete graph substructures for degree sequences

Abstract

Given a graphic degree sequence D, let χ(D) (respectively ω(D), h(D), and H(D)) denote the maximum value of the chromatic number (respectively, the size of the largest clique, largest clique subdivision, and largest clique minor) taken over all simple graphs whose degree sequence is D. It is proved that χ(D)≤h(D). Moreover, it is shown that a subdivision of a clique of order χ(D) exists where each edge is subdivided at most once and the set of all subdivided edges forms a collection of disjoint stars. This bound is an analogue of the Hajós Conjecture for degree sequences and, in particular, settles a conjecture of Neil Robertson that degree sequences satisfy the bound χ(D) ≤ H(D) (which is related to the Hadwiger Conjecture). It is also proved that χ(D) ≤ 6/5 ω(D)+ 3/5 and that χ(D) ≤ 4/5 ω(D) + 1/5 Δ(D)+1, where Δ(D) denotes the maximum degree in D. The latter inequality is related to a conjecture of Bruce Reed bounding the chromatic number by a convex combination of the clique number and the maximum degree. All derived inequalities are best possible

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Supported in part through a postdoctoral position at Simon Fraser University and by the grant GA201/09/0197 of Czech Science Foundation.

On leave from: IMFM & FMF, Department of Mathematics, University of Ljubljana, Ljubljana, Slovenia.

Supported in part by an NSERC Discovery Grant (Canada), by the Canada Research Chair program, and by the Research Grant P1-0297 of ARRS (Slovenia).

On leave from: Institute of Theoretical Informatics, Charles University, Prague, Czech Republic.

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Dvořák, Z., Mohar, B. Chromatic number and complete graph substructures for degree sequences. Combinatorica 33, 513–529 (2013). https://doi.org/10.1007/s00493-013-2649-z

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Mathematics Subject Classification (2010)

  • 05C15
  • 05C07