Chromatic polynomials of complementary (n,k)-clique pairs

TL;DR

This paper studies pairs of graphs formed by two cliques connected in a complementary manner, revealing a precise relationship between their chromatic polynomials, splitting fields, and acyclic orientations, thus advancing understanding of graph coloring properties.

Contribution

It introduces a new class of graph pairs with complementary clique-bridge structures and establishes exact relations between their chromatic polynomials and related properties.

Findings

Graphs in the pair have the same splitting field.

Number of acyclic orientations relates to the other's proper colorings.

Chromatic polynomials are intricately connected through complementarity.

Abstract

We introduce a class of pairs of graphs consisting of two cliques joined by an arbitrary number of edges. The members of a pair have the property that the clique-bridging edge-set of one graph is the complement of that of the other. We prove a precise relation between the chromatic polynomials of the graphs in such a pair, showing that they have the same splitting field, and that the number of acyclic orientations of each graph is determined by the number of proper vertex-colourings of the other.