A counterexample to Montgomery's conjecture on dynamic colourings of regular graphs

TL;DR

This paper constructs specific regular graphs demonstrating that Montgomery's conjecture, which proposed a tight bound on the dynamic colouring number relative to the chromatic number, is false.

Contribution

It provides a counterexample for all integers n ≥ 2, disproving Montgomery's conjecture on the upper bound of dynamic colouring numbers.

Findings

Counterexamples for all n ≥ 2 with χ(G)=n and χ₂(G)=2n

Disproves Montgomery's conjecture on dynamic colourings

Shows the upper bound χ₂(G) ≤ 2χ(G) is sharp

Abstract

A \emph{dynamic colouring} of a graph is a proper colouring in which no neighbourhood of a non-leaf vertex is monochromatic. The \emph{dynamic colouring number} $\chi_2(G)$ of a graph $G$ is the least number of colours needed for a dynamic colouring of $G$. Montgomery conjectured that $\chi_2(G) \leq \chi(G) + 2$ for all regular graphs $G$, which would significantly improve the best current upper bound $\chi_2(G) \leq 2\chi(G)$. In this note, however, we show that this last upper bound is sharp by constructing, for every integer $n \geq 2$, a regular graph $G$ with $\chi(G) = n$ but $\chi_2(G) = 2n$. In particular, this disproves Montgomery's conjecture.

Equations

$$\gamma (G):=\min \{\chi (G[D]):D\text{ a total dominating set of }G\}.$$

Full text

Montgomery conjectured that $\chi_{2}(G)\leq\chi(G)+2$ for all regular graphs $G$, which would significantly improve the best current upper bound $\chi_{2}(G)\leq 2\chi(G)$. In this note, however, we show that this last upper bound is sharp by constructing, for every integer $n\geq 2$, a regular graph $G$ with $\chi(G)=n$ but $\chi_{2}(G)=2n$. In particular, this disproves Montgomery’s conjecture.

The second author was supported by the Alexander von Humboldt Foundation.

1. Introduction

A proper $n$-colouring of a graph $G$ is a map $c\colon V(G)\rightarrow[n]$ such that there is no edge $uv$ with $c(u)=c(v)$. An $r$-dynamic $n$-colouring is a proper $n$-colouring $c$ of $G$ such that for each vertex $v\in V(G)$ at least $\min\{r,d(v)\}$ colours are used on $N(v)$. The $r$-dynamic chromatic number $\chi_{r}(G)$, introduced by Montgomery [13], is the smallest $n$ such that $G$ has an $r$-dynamic $n$-colouring. Problems on $r$-dynamic colourings of graphs are an active area of research in graph theory, with many recent papers investigating properties of $r$-dynamic colourings, e.g. [2, 6, 10, 11, 12].

Of particular interest is the 2-dynamic chromatic number, which is often simply called the dynamic chromatic number. Note that $\chi_{2}(G)-\chi(G)$ can be arbitrarily large, which can be seen by considering subdivisions of complete graphs. However, in the case of regular graphs, Montgomery [13] conjectured the following:

Conjecture 1** (Montgomery’s Conjecture).**

If $G$ is a regular graph, then $\chi_{2}(G)\leq\chi(G)+2$.

Ahadi, Akbari, Deghan, and Ghanbari [1] conjectured further that if $\chi(G)\geq 4$ then $\chi(G)=\chi_{2}(G)$. However Alishahi [5] disproved the stronger conjecture by constructing graphs $G$ with $\chi(G)=n$ such that $\chi_{2}(G)\geq\chi(G)+1$ for each $n\geq 2$.

There are several results showing that Conjecture 1 holds for certain classes of regular graphs. For example, Montgomery [13] showed that Conjecture 1 holds for claw-free graphs, and Akbari, Ghanbari and Jahanbekam [3] showed that Conjecture 1 holds for $d$-regular bipartite graphs with $d\geq 4$. Alishahi [5] verified Conjecture 1 if $G$ has diameter at most $2$ and chromatic number at least $4$.

There are also results bounding the dynamic chromatic number of regular graphs in terms of other invariants of the graph. For example, Dehghan, and Ahadi [8] showed that if $G$ is regular, then $\chi_{2}(G)\leq\chi(G)+2\log{\alpha(G)}+3$, where $\alpha$ is the independence number of the graph. Alishahi [4] showed that if $G$ is $d$-regular, then $\chi_{2}(G)\leq\chi(G)+14.06\log{d}+1$, and Taherkhani [17] improved this bound to $\chi_{2}(G)\leq\chi(G)+5.437\log{d}+2$. The best general bound for $\chi_{2}(G)$ depending only on $\chi(G)$ was given by Alishahi [4] who showed that $\chi_{2}(G)\leq 2\chi(G)$ for all regular graphs $G$.

in this paper we show that for every value of $\chi(G)$ there exists a graph for which this inequality is sharp, giving an infinite family of counterexamples to Montgomery’s Conjecture.

Theorem 1.1**.**

For every natural number $n\geq 2$, there exists a regular graph $G_{n}$ with $\chi(G_{n})=n$ and $\chi_{2}(G_{n})=2\chi(G_{n})$.

In general, the $r$-dynamic chromatic number of regular graphs cannot be bounded by $r\chi_{r}(G)$ as was shown by Jahanbekam, Kim, O, and West [9]. They showed that if $d=r$ then for infinitely many values of $r$ there exists a $d$-regular graph $G$ such that $\chi_{r}(G)\geq r^{1.377}\chi(G)$. However, they also showed that $\chi_{r}(G)\leq r\chi(G)$ for $d\geq\left(3+o(1)\right)r\log{r}$. Here we show that this bound is sharp even for arbitrarily large values of $d$.

Theorem 1.2**.**

For all natural numbers $r,n,\delta\geq 2$, there exists a $d$-regular graph $G=G(r,n,\delta)$ with $d>\delta$, $\chi(G)=n$ and $\chi_{r}(G)=r\chi(G)$.

The dynamic chromatic number also has connections to the chromatic number of total dominating sets, which was observed by Alishahi [4]. We say a set $D\subset V(G)$ is a total dominating set of $G$ if every vertex in $V(G)$ is adjacent to some vertex of $D$. Let us write

[TABLE]

In the 1970s, Berge (unpublished, see [7]) and independently Payan [14] conjectured that every regular graph contains two disjoint maximal independent sets. Payan [15, 16] showed that this is equivalent to $\gamma(G)=2$ for every regular graph $G$ and disproved the conjecture by constructing regular graphs $G_{p}$ with $\gamma(G_{p})>p$ for every $p\in\mathbb{N}$. His graphs have the property that $\chi(G_{p})=4p+1$ and $\gamma(G_{p})=p+1$. The graphs we construct here are new examples showing that $\gamma(G)$ is unbounded for regular graphs. Moreover, our graphs have the even stronger property that $\chi(G)=\gamma(G)$.

Theorem 1.3**.**

For every natural number $n\geq 2$, there exists a regular graph $G_{n}$ with $\chi(G_{n})=n=\gamma(G_{n})$.

2. Construction of the counterexamples

Our aim is to build, for fixed natural numbers $r,n,\delta\geq 2$, a $d$-regular graph $G:=G(r,n,\delta)$ with $d>\delta$ and $\chi(G)=n$ but $\chi_{r}(G)=rn$ and $\gamma(G)=n$. The graph $G$ will consist of a disjoint union of many complete $n$-partite graphs, together with some supplementary vertices whose neighbourhoods are carefully chosen subsets of that disjoint union.

More formally, letting $m:=\max\{\binom{rn-1}{r-1}n^{2},\delta\}$ and $N:=\binom{m-1}{n-1}$, we take a vertex $v_{i,j,k}$ for each $i\in[n]$, $j\in[N]$ and $k\in[m]$ and we add an edge from $v_{i,j,k}$ to $v_{i^{\prime},j^{\prime},k^{\prime}}$ whenever $i\neq i^{\prime}$ and $k=k^{\prime}$. Additionally we take vertices $s_{i,X}$ for every $i\in[n]$ and every $X\subseteq[m]$ of size $n$, and we join $s_{i,X}$ to $v_{i^{\prime},j,k}$ whenever $i=i^{\prime}$ and $k\in X$ (see Fig 1). We call the graph with these vertices and edges $G$.

Lemma 2.1**.**

$G$* is $(nN)$-regular.*

Proof.

It is clear that vertices of the form $s_{i,X}$ have degree $nN$. Now we consider the neighbours of a vertex of the form $v_{i,j,k}$. This vertex has $(n-1)N$ neighbours of the form $v_{i^{\prime},j^{\prime},k^{\prime}}$, and it has $N$ neighbours of the form $s_{i,X}$ since there are $\binom{m-1}{n-1}$ subsets $X$ of $[m]$ of size $n$ containing $k$. So in total it has $nN$ neighbours. ∎

Lemma 2.2**.**

$\chi(G)=n$.

Proof.

The chromatic number is at least $n$, since the subgraph on the $v_{i,1,1}$ with $i\in[n]$ is complete. But $G$ can be properly coloured with $n$ colours by assigning each vertex $v_{i,j,k}$ the colour $i$ and each vertex $s_{i,X}$ a colour other than $i$. ∎

Lemma 2.3**.**

$\chi_{r}(G)=rn$.

Proof.

We can $r$-dynamically colour $G$ with $rn$ colours by assigning each vertex $v_{i,j,k}$ with $j<r$ the colour $nj+i$, each vertex $v_{i,j,k}$ with $j\geq r$ the colour $i$, and each vertex $s_{i,X}$ a colour which is not congruent to $i$ modulo $n$.

Now consider a proper colouring of $G$ with at most $rn-1$ colours. We shall show that this colouring is not $r$-dynamic. For each $k\in[m]$ and each colour $c$, there is at most one $i\in[n]$ such that for some $j$ the vertex $v_{i,j,k}$ has colour $c$. Since there are only $rn-1$ colours, there must be some $i_{k}\in[n]$ and some set $C_{k}$ of colours of size at most $r-1$ such that all $v_{i_{k},j,k}$ are coloured with a colour from $C_{k}$. By the pigeonhole principle, since $m>(n-1)n\binom{rn-1}{r-1}$, there is a set $X\subseteq[m]$ of size $n$ such that all $i_{k}$ with $k\in X$ take some common value $i$ and all $C_{k}$ with $k\in X$ take some common value $C$. But then all neighbours of $s_{i,X}$ are coloured with colours from $C$, and so the colouring is not $r$-dynamic. ∎

Lemma 2.4**.**

$\gamma(G)=n$.

Proof.

Consider a set $S\subseteq V(G)$ with $\chi(G[S])\leq n-1$. For each $k\in[m]$ there exists at least one $i_{k}\in[n]$ such that $S$ contains none of the $v_{i_{k},j,k}$. Since $m>(n-1)n$, by the pigeonhole principle, there exists a set $X\subseteq[m]$ of size $n$ such that all $i_{k}$ with $k\in X$ take some common value $i$. Now none of the neighbours of $s_{i,X}$ are in $S$, so $S$ is not a total dominating set. ∎