Maximum diameter of $3$- and $4$-colorable graphs

Éva Czabarka; Stephen J. Smith; László Székely

Maximum diameter of $3$- and $4$-colorable graphs

Combinatorics 2021-09-29 v1

Authors: Éva Czabarka , Stephen J. Smith , László Székely

Abstract

P. Erd\H{o}s, J. Pach, R. Pollack, and Z. Tuza [J. Combin. Theory, B 47 (1989), 279--285] made conjectures for the maximum diameter of connected graphs without a complete subgraph $K_{k+1}$ , which have order $n$ and minimum degree $\delta$ . Settling a weaker version of a problem, by strengthening the $K_{k+1}$ -free condition to $k$ -colorable, we solve the problem for $k=3$ and $k=4$ using a unified linear programming duality approach. The case $k=4$ is a substantial simplification of the result of \'E. Czabarka, P. Dankelmann, and L. A. Sz\'ekely [Europ. J. Comb., 30 (2009), 1082--1089].

Keywords

graph coloring rainbow coloring graph theory

Cite

@article{arxiv.2109.13887,
  title  = {Maximum diameter of $3$- and $4$-colorable graphs},
  author = {Éva Czabarka and Stephen J. Smith and László Székely},
  journal= {arXiv preprint arXiv:2109.13887},
  year   = {2021}
}

Comments

8 pages. arXiv admin note: text overlap with arXiv:2009.02611

Maximum diameter of $3$- and $4$-colorable graphs

Abstract

Keywords

Cite

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