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The chromatic index of strongly regular graphs

Sebastian M. Cioaba, Krystal Guo, Willem H. Haemers

TL;DR

The paper investigates the edge-chromatic number $\chi'(G)$ for strongly regular graphs (SRGs), focusing on when SRGs are class 1 (i.e., $\chi'(G)=k$ for a $k$-regular graph). It combines the Ferber–Jain asymptotic criterion with constructive partition-based arguments and case analyses of SRG families (Latin square graphs, triangular graphs, Steiner 2-design block graphs) to establish broad class 1 results and several explicit exceptions. It also conducts an extensive computational verification for primitive SRGs of even order with $k\le 18$ and their complements, finding that all are class 1 except the Petersen graph, and provides public code for the search. The results support a conjecture that every connected SRG of even order is class 1, illustrating when SRGs admit $1$-factorizations and highlighting practical algorithms to obtain them.

Abstract

We determine (partly by computer search) the chromatic index (edge-chromatic number) of many strongly regular graphs (SRGs), including the SRGs of degree $k \leq 18$ and their complements, the Latin square graphs and their complements, and the triangular graphs and their complements. Moreover, using a recent result of Ferber and Jain it is shown that an SRG of even order $n$, which is not the block graph of a Steiner 2-design or its complement, has chromatic index $k$, when $n$ is big enough. Except for the Petersen graph, all investigated connected SRGs of even order have chromatic index equal to their degree, i.e., they are class 1, and we conjecture that this is the case for all connected SRGs of even order.

The chromatic index of strongly regular graphs

TL;DR

The paper investigates the edge-chromatic number for strongly regular graphs (SRGs), focusing on when SRGs are class 1 (i.e., for a -regular graph). It combines the Ferber–Jain asymptotic criterion with constructive partition-based arguments and case analyses of SRG families (Latin square graphs, triangular graphs, Steiner 2-design block graphs) to establish broad class 1 results and several explicit exceptions. It also conducts an extensive computational verification for primitive SRGs of even order with and their complements, finding that all are class 1 except the Petersen graph, and provides public code for the search. The results support a conjecture that every connected SRG of even order is class 1, illustrating when SRGs admit -factorizations and highlighting practical algorithms to obtain them.

Abstract

We determine (partly by computer search) the chromatic index (edge-chromatic number) of many strongly regular graphs (SRGs), including the SRGs of degree and their complements, the Latin square graphs and their complements, and the triangular graphs and their complements. Moreover, using a recent result of Ferber and Jain it is shown that an SRG of even order , which is not the block graph of a Steiner 2-design or its complement, has chromatic index , when is big enough. Except for the Petersen graph, all investigated connected SRGs of even order have chromatic index equal to their degree, i.e., they are class 1, and we conjecture that this is the case for all connected SRGs of even order.

Paper Structure

This paper contains 4 sections, 10 theorems, 3 equations, 1 table.

Table of Contents

  1. Introduction
  2. Sufficient conditions for being class 1
  3. Asymptotic results
  4. SRGs of degree at most $18$

Key Result

Theorem 1.1

There exist universal constants $n_0$ and $k_0$, such that the following holds. If $G$ is a connected $k$-regular graph of even order $n$ with eigenvalues $k=\vartheta_1>\vartheta_2\geq\ldots\geq\vartheta_n$, and $n>n_0$, $k>k_0$ and $\max\{\vartheta_2,-\vartheta_n\}<k^{0.9}$, then $G$ is class $1$.

Theorems & Definitions (14)

  • Theorem 1.1
  • Conjecture 1.2
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • Corollary 2.4
  • Theorem 3.1
  • proof
  • ...and 4 more