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Bivariate Chromatic Polynomials of Mixed Graphs

Matthias Beck, Sampada Kolhatkar

Abstract

The bivariate chromatic polynomial $χ_G(x,y)$ of a graph $G = (V, E)$, introduced by Dohmen-Pönitz-Tittmann (2003), counts all $x$-colorings of $G$ such that adjacent vertices get different colors if they are $\le y$. We extend this notion to mixed graphs, which have both directed and undirected edges. Our main result is a decomposition formula which expresses $χ_G(x,y)$ as a sum of bivariate order polynomials (Beck-Farahmand-Karunaratne-Zuniga Ruiz 2020), and a combinatorial reciprocity theorem for $χ_G(x,y)$.

Bivariate Chromatic Polynomials of Mixed Graphs

Abstract

The bivariate chromatic polynomial of a graph , introduced by Dohmen-Pönitz-Tittmann (2003), counts all -colorings of such that adjacent vertices get different colors if they are . We extend this notion to mixed graphs, which have both directed and undirected edges. Our main result is a decomposition formula which expresses as a sum of bivariate order polynomials (Beck-Farahmand-Karunaratne-Zuniga Ruiz 2020), and a combinatorial reciprocity theorem for .
Paper Structure (5 sections, 5 theorems, 24 equations, 3 figures)

This paper contains 5 sections, 5 theorems, 24 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Chromatic and (Bicolored) Order Polynomials
  3. Deletion--Contraction
  4. Decomposition into Order Polynomials
  5. Reciprocity

Key Result

Proposition 1

If $G= \left(V,E,A\right)$ is a mixed graph and $e \in E$ is an edge, then If $a = \overrightarrow{uv} \in A$ is an arc, then

Figures (3)

  • Figure 1: A mixed graph $G$.
  • Figure 2: A mixed graph, one of its flat and the associated undirected graph.
  • Figure 3: Acyclic orientations of contractions of $G$.

Theorems & Definitions (12)

  • Definition
  • Example 1
  • Example 2
  • Proposition 1
  • proof : of \ref{['eq:delcontrarc']}
  • Theorem 1
  • proof
  • Corollary 1
  • Example 3
  • Theorem 2
  • ...and 2 more