An improved bound in Vizing's conjecture

TL;DR

This paper improves a lower bound related to Vizing's conjecture on domination numbers in Cartesian product graphs, strengthening previous results with a tighter inequality.

Contribution

The authors present a new lower bound for the domination number of the Cartesian product of graphs, advancing the understanding of Vizing's conjecture.

Findings

Established a new lower bound involving max of domination numbers

Improved previous bounds by Suen and Tarr

Contributed to the theoretical understanding of graph domination

Abstract

A well-known conjecture of Vizing is that $\gamma(G \square H) \ge \gamma(G)\gamma(H)$ for any pair of graphs $G, H$, where $\gamma$ is the domination number and $G \square H$ is the Cartesian product of $G$ and $H$. Suen and Tarr, improving a result of Clark and Suen, showed $\gamma(G \square H) \ge \frac{1}{2}\gamma(G)\gamma(H) + \frac{1}{2}\min(\gamma(G),\gamma(H))$. We further improve their result by showing $\gamma(G \square H) \ge \frac{1}{2}\gamma(G)\gamma(H) + \frac{1}{2}\max(\gamma(G),\gamma(H)).$

Equations

$$\gamma (G\square H)\ge \frac{1}{2}\gamma (G)\gamma (H)+\frac{1}{2}\max (\gamma (G),\gamma (H)).$$

$$|V(H)-N_H[P_i]|\ge \gamma (H)-|P_i|.$$

$$C=\{(i,v)\mid {\pi }_i\times \{v\}\subset N_{G\square H}[Q_v]\}.$$

$$\begin{array}{rl}|C|=& \sum _{i=1}^k|L_i|\ge \sum _{i=1}^k(|V(H)-N_H[P_i]|+|T_i|)\\ & \ge \sum _{i=1}^k(\gamma (H)-|P_i|+1)\ge k\gamma (H)-|D|+k.\end{array}$$

$$U^{\prime }=\{u\mid (u,v)\in Q_v\}\cup \{u_j\mid (j,v)\notin R_v\}$$

$$|C|\le \sum _{v\in V(H)}|R_v|\le \sum _{v\in V(H)}|Q_v|=|D|.$$

$$2|D|\ge k\gamma (H)+k\ge \gamma (G)\gamma (H)+\gamma (G)=\gamma (G)\gamma (H)+\max (\gamma (G),\gamma (H)),$$

$$\begin{array}{rl}|C|=& \sum _{i=1}^k|L_i|\ge \sum _{i=1}^k(|V(H)-N_H[P_i]|+|T_i|)\\ & \ge \sum _{i=1}^k(\gamma (H)-|P_i|+|D_i|)\ge \gamma (G)\gamma (H).\end{array}$$

Full text

This research was partly supported by the New-England Fund, Technion.

For a simple graph $G=(V(G),E(G))$ and a vertex $v\in V(G)$, denote by $N_{G}[v]$ the neighborhood of $v$ in $G$, namely the set $\{v\}\cup\{u\in V(G)\mid uv\in E(G)\}$. We say that $D\subseteq V(G)$ dominates $G$ if $V(G)=\bigcup_{v\in D}N_{G}[v]$. The domination number $\gamma(G)$ is the minimal size of a set dominating $G$. The Cartesian product $G\square H$ of a pair of graphs $G,H,$ is the graph whose vertex set is $V(G)\times V(H)$, and whose edge set consists of pairs $(u,v)(u^{\prime},v^{\prime})$ in which either $u=u^{\prime}$ and $vv^{\prime}\in E(H)$ or $v=v^{\prime}$ and $uu^{\prime}\in E(G)$.

In 1963 Vizing [7] conjectured that $\gamma(G\square H)\geq\gamma(G)\gamma(H)$. Although proven for certain families of graphs and for all pairs of graphs $G,H$ for which $\gamma(G)\leq 3$, this conjecture is still wide open. For surveys on Vizing’s conjecture and recent results related to it see [1, 4].

In 2000 Clark and Suen [2] showed that for every pair of graphs $G,H$ we have $\gamma(G\square H)\geq\frac{1}{2}\gamma(G)\gamma(H)$. Suen and Tarr [6] then improved this result by showing $\gamma(G\square H)\geq\frac{1}{2}\gamma(G)\gamma(H)+\min(\gamma(G),\gamma(H)).$ Here we further improve these results, establishing $\gamma(G\square H)\geq\frac{1}{2}\gamma(G)\gamma(H)+\frac{1}{2}\max(\gamma(G),\gamma(H)).$. In particular, our proof is simpler the the proof of Suen and Tarr in [6].

[TABLE]

Form a partition $\pi_{i},~{}i=1,\dots,k,$ of $V(G)$, so that $u_{i}\in\pi_{i}$ and $\pi_{i}\subseteq N_{G}[u_{i}]$ for all $i$. This induces a partition $D_{i},~{}i=1,\dots,k,$ of $D$, where $D_{i}=(\pi_{i}\times V(H))\cap D$. Let $P_{i}$ be the projection of $D_{i}$ onto $H$. Observe that the set $P_{i}\cup(V(H)-N_{H}[P_{i}])$ dominates $H$, and hence for all $i$,

[TABLE]

For $v\in V(H)$ let $Q_{v}=D\cap(V(G)\times\{v\})$ and define

[TABLE]

Set $L_{i}=\{(i,v)\in C\mid v\in V(H)\}$ and $R_{v}=\{(i,v)\in C\mid 1\leq i\leq k\}.$ Then we have, $|C|=\sum_{i=1}^{k}|L_{i}|=\sum_{v\in H}|R_{v}|$.

Observe that if $v\in(V(H)-N_{H}[P_{i}])\cup T_{i}$, then the vertices in $\pi_{i}\times\{v\}$ are dominated by the vertices in $Q_{v}$ and therefore $(i,v)\in L_{i}$. Since the sets $V(H)-N_{H}[P_{i}]$ and $T_{i}$ are disjoint, this implies that $|L_{i}|\geq|V(H)-N_{H}[P_{i}]|+|T_{i}|$.

Therefore,

[TABLE]

We further claim that $|R_{v}|\leq|Q_{v}|$. Indeed, if not then the set

[TABLE]

is a dominating set of $G$ of cardinality $|U^{\prime}|<|U|$. Moreover, since the projection of $Q_{v}$ onto $G$ is a subset of $Q$, we have that $U^{\prime}\subset U$, and thus we obtain a contradiction to the minimality of $U$.

Thus we have

[TABLE]

Combining (1) and (2) together we get

[TABLE]

which concludes the proof of the theorem. ∎

[TABLE]

Combining this with Equation (2) we get the result.

Unfortunately, there exist pairs of graphs $G,H$ for which such $D$ does not exist. An example of such a pair is $G=H=P_{4}$, where $P_{4}$ is a path with $4$ vertices.**

Acknowledgement

The author is grateful to Ron Aharoni for many helpful discussions.