Recursive Formulae for the Chromatic Polynomials of Complete \( r \)-Uniform Mixed Interval Hypergraphs

Abstract

A graph \(G\) is said to be equitably \(k\)-colorable if the vertex set of \(G\) can be divided into \(k\) independent sets for which any two sets differ in size at most one. The equitable chromatic number of \(G\), \(\chi_{=}(G)\), is the minimum \(k\) for which \(G\) is equitably \(k\)-colorable. The equitable chromatic threshold of \(G\), \(\chi_m^*(G)\), is the minimum \(k\) for which \(G\) is equitably \(k’\)-colorable for all \(k’ \geq k\). In this paper, the exact values of \(\chi_m^*(P_{n’,2} \square K_{m,n})\) and \(\chi_{=}(P_{n’,m} \square K_{m,n})\) are obtained except that \(3 \leq \xi_m^*(P_{5,2} \square K_{m,n}) = \chi_{=}(P_{s,m} \square K_{m,n}) \leq 4\) when \(m+n \geq 3\min\{m,n\} + 2\) or \(m+n < 3\min\{m,n\} – 2\).