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arXiv · 2407.00883

Chromatic polynomials of signed graphs and dominating-vertex deletion formulae

Gary R. W. Greaves·Jeven Syatriadi·Charissa I. Utomo
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Abstract

We exhibit non-switching-isomorphic signed graphs that share a common underlying graph and common chromatic polynomials, thereby answering a question posed by Zaslavsky. For various joins of all-positive or all-negative signed complete graphs, we derive a closed-form expression for their chromatic polynomials. As a generalisation of the chromatic polynomials for a signed graph, we introduce a new pair of bivariate chromatic polynomials. We establish recursive dominating-vertex deletion formulae for these bivariate chromatic polynomials. Finally, we show that for certain families of signed threshold graphs, isomorphism is equivalent to the equality of bivariate chromatic polynomials.

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Gary R. W. Greaves, Jeven Syatriadi, Charissa I. Utomo. 2024-07-01. Chromatic polynomials of signed graphs and dominating-vertex deletion formulae. https://arxiv.org/abs/2407.00883

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