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arXiv · 1903.03821

Equality cases for a bound on the chromatic number

Boon Suan Ho·Joel Junyao Tan·Xiaorui Zhang
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Abstract

It is known that the inequality $$ \frac{χ(G)(χ(G)-1)}{2} + |V| - χ(G) \leq |E|$$ holds for all connected graphs, where $χ(G)$ 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.

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Boon Suan Ho, Joel Junyao Tan, Xiaorui Zhang. 2019-03-09. Equality cases for a bound on the chromatic number. https://arxiv.org/abs/1903.03821

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