On the $d$-transversal number of cylindrical and toroidal grids

Abstract

For a positive integer $d$, a $d$-transversal set of a graph $G$ is an edge subset $T\subseteq E(G)$ such that $|T\cap M|\geq d$ for every maximum matching $M$ of $G$. The $d$-transversal number of $G$, denoted by $τ_d(G)$, is the minimum cardinality of a $d$-transversal set in $G$. It is NP-complete to determine the $d$-transversal number of a bipartite graph for any fixed $d\geq 1$. Ries et al. (Discrete Math. 310 (2010) 132-146) established the $d$-transversal number of rectangular grids $P_m\square P_n$. In this paper, we consider cylindrical grids $P_m\square C_n$ and toroidal grids $C_m\square C_n$. We derive explicit expressions for the $d$-transversal numbers of $P_m\square C_n$ for $m\geq 1$ and even $n\geq 4$, or even $m\geq 2$ and $n=3$, and of $C_m\square C_n$ with even order, for $1\leq d\leq \frac{mn}{2}$. For the other cases we obtain explicit expressions or bounds for their $d$-transversal numbers.

Figures (9)

• Figure 1: (a)$P_3\square C_4$; (b)$C_3\square C_4$.
• Figure 2: A minimum $2$-transversal set $T$ (thick edges) of $G$.
• Figure 3: (a)Perfect matchings $M_1$ and $M_2$ of $P_4\square C_6$; (b)Perfect matchings $M_3$ and $M_4$ of $P_4\square C_6$.
• Figure 4: 4-partition of $P_6\square C_7$.
• Figure 5: Perfect matchings $M^2_4$ (left) and $M^3_4$ (right) of $P_6\square C_7$.

Theorems & Definitions (54)

• Theorem 1.1
• Theorem 1.2
• Proposition 2.1
• Proposition 2.2
• proof
• Proposition 2.3
• proof
• Lemma 3.1
• proof
• Lemma 3.2