Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the distinguishing chromatic number in hereditary graph classes

Christoph Brause, Rafał Kalinowski, Monika Pilśniak, Ingo Schiemeyer

TL;DR

This work investigates χ_D(G) in hereditary graph classes defined by forbidding small induced subgraphs, deriving tight upper bounds such as χ_D(G) ≤ Δ(G)+1 for several C4-free and related classes. It develops structural tools around simplicial vertices and dominating modules to design distinguishing colourings, and it leverages line-graph theory and Whitney’s isomorphism to handle claw-diamond-free graphs via edge-distinguishing colourings. Equality cases are carefully characterized (e.g., C6, C5, symmetric trees, and specific joining constructions), highlighting when tighter colourings are possible than the general bound χ_D(G) ≤ 2Δ(G). The results unify and extend prior bounds, offering concrete guidance for distinguishing colourings in sparse hereditary graph classes and related line-graph scenarios.

Abstract

The distinguishing chromatic number of a graph $G$, denoted $χ_D(G)$, is the minimum number of colours in a proper vertex colouring of $G$ that is preserved by the identity automorphism only. Collins and Trenk proved that $χ_D(G)\le 2Δ(G)$ for any connected graph $G$, and the equality holds for complete balanced bipartite graphs $K_{p,p}$ and for $C_6$. In this paper, we show that the upper bound on $χ_D(G)$ can be substantially reduced if we forbid some small graphs as induced subgraphs of $G$, that is, we study the distinguishing chromatic number in some hereditary graph classes.

On the distinguishing chromatic number in hereditary graph classes

TL;DR

This work investigates χ_D(G) in hereditary graph classes defined by forbidding small induced subgraphs, deriving tight upper bounds such as χ_D(G) ≤ Δ(G)+1 for several C4-free and related classes. It develops structural tools around simplicial vertices and dominating modules to design distinguishing colourings, and it leverages line-graph theory and Whitney’s isomorphism to handle claw-diamond-free graphs via edge-distinguishing colourings. Equality cases are carefully characterized (e.g., C6, C5, symmetric trees, and specific joining constructions), highlighting when tighter colourings are possible than the general bound χ_D(G) ≤ 2Δ(G). The results unify and extend prior bounds, offering concrete guidance for distinguishing colourings in sparse hereditary graph classes and related line-graph scenarios.

Abstract

The distinguishing chromatic number of a graph , denoted , is the minimum number of colours in a proper vertex colouring of that is preserved by the identity automorphism only. Collins and Trenk proved that for any connected graph , and the equality holds for complete balanced bipartite graphs and for . In this paper, we show that the upper bound on can be substantially reduced if we forbid some small graphs as induced subgraphs of , that is, we study the distinguishing chromatic number in some hereditary graph classes.

Paper Structure

This paper contains 5 sections, 17 theorems, 29 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Dealing with simplicial vertices
  3. $C_4$-free graphs and $2K_2$-free graphs
  4. $claw$-free graphs
  5. ($claw,diamond$)-free graphs

Key Result

Theorem 1

(Collins, Trenk CT) If $G$ is a connected graph with maximum degree $\Delta$, then with equality if and only if $G\cong K_{\Delta,\Delta}$ or $G\cong C_6$.

Figures (4)

  • Figure 1: An example of a symmetric tree $T_s$
  • Figure 3: An example of a symmetric graph $T_B$
  • Figure 4: Graph $G=3K_1+K_4$
  • Figure 5: The line graph $L(K_{1,3})$

Theorems & Definitions (29)

  • Theorem 1
  • Theorem 2
  • Proposition 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Theorem 6
  • proof
  • ...and 19 more