Chromatic Polynomials of Signed Book Graphs

TL;DR

This paper investigates the chromatic properties of signed book graphs, providing formulas for their chromatic polynomials and classifying their chromatic numbers, with a focus on switching non-isomorphic cases.

Contribution

It establishes the count of switching non-isomorphic signed book graphs and derives explicit chromatic polynomial formulas for these graphs.

Findings

Number of switching non-isomorphic signed B(m,n) graphs is n+1

Chromatic number of signed B(m,n) is either 2 or 3

Explicit formulas for chromatic and zero-free chromatic polynomials

Abstract

For $m \geq 3$ and $n \geq 1$, the $m$-cycle book graph $B(m,n)$ consists of $n$ copies of the cycle $C_m$ with one common edge. In this paper, we prove that (a) the number of switching non-isomorphic signed $B(m,n)$ is $n+1$, and (b) the chromatic number of a signed $B(m,n)$ is either 2 or 3. We also obtain explicit formulas for the chromatic polynomials and the zero-free chromatic polynomials of switching non-isomorphic signed book graphs.