Equality cases for a bound on the chromatic number

Boon Suan Ho; Joel Junyao Tan; Xiaorui Zhang

Equality cases for a bound on the chromatic number

Combinatorics 2019-03-12 v1

Authors: Boon Suan Ho , Joel Junyao Tan , Xiaorui Zhang

Abstract

It is known that the inequality $\frac{\chi(G)(\chi(G)-1)}{2} + |V| - \chi(G) \leq |E|$ holds for all connected graphs, where $\chi(G)$ denotes the chromatic number of $G$ . We prove that equality holds whenever the graph consists of a complete graph or an odd cycle, together with finitely many trees attached to its vertices.

Keywords

graph coloring graph theory graph pebbling

Cite

@article{arxiv.1903.03821,
  title  = {Equality cases for a bound on the chromatic number},
  author = {Boon Suan Ho and Joel Junyao Tan and Xiaorui Zhang},
  journal= {arXiv preprint arXiv:1903.03821},
  year   = {2019}
}

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5 pages

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