STAR MATCHING AND DISTANCE TWO LABELLING

Abstract

This paper first introduces a new graph parameter. Let $t$ be a positive integer. A $t$-star-matching of a graph $G$ is a collection of mutually vertex disjoint subgraphs $K_{1,i}$ of $G$ with $1 \leq i \leq t$. The $t$-star-matching number, denoted by $SM_t(G)$, is the maximum number of vertices covered by a $t$-star-matching of $G$. Clearly $SM_1(G)/2$ is the edge independence number of $G$.

An $L(2,1)$-labelling of a graph $G$ is an assignment of nonnegative integers to the vertices of $G$ such that vertices at distance at most two get different numbers and adjacent vertices get numbers which are at least two apart. The $L(2,1)$-labelling number of a graph $G$ is the minimum range of labels over all $L(2,1)$-labellings. If we require the assignment to be one-to-one, then similarly as above we can define the $L'(2,1)$-labelling and the $L'(2,1)$-labelling number of a graph $G$. Given a graph $G$, the path covering number of $G$, denoted by $p_v(G)$, is the smallest number of vertex-disjoint paths covering $V(G)$. By $G^c$ we denote the complement graph of $G$.

In this paper, we design a polynomial time algorithm to compute $SM_t(G)$ for any graph $G$ and any integer $t\geq 2$ and studies the properties of $t$-star-matchings of a graph $G$. For any graph $G$, we determine the path covering numbers of $(\mu(G))^c$ and $(G\times \hat{K}_2)^c$ in terms of $SM_4(G^c)$, and the $L'(2,1)$-labelling umbers of $\mu(G)$ and $G\times \hat{K}_2$ in terms of $SM_4(G^c)$, where $\mu(G)$ is the Mycielskian of $G$ and $G\times \hat{K}_2$ is the direct product of $G$ and $\hat{K}_2$ ($\hat{K}_2$ is a graph obtained from $K_2$ by adding a loop on one of its vertices). Our results imply that the path covering numbers of $(\mu(G))^c$ and $(G\times \hat{K}_2)^c$, the $L'(2,1)$-labelling umbers of $\mu(G)$ and $G\times \hat{K}_2$ can be computed in polynomial time for any graph $G$. So, for any graph $G$, it is polynomial-time solvable to determine whether $(\mu(G))^c$ and $(G\times \hat{K}_2)^c$ has a Hamiltonian path. And consequently, for any graph $G=(V,E)$, it is polynomially solvable to determine whether $\lambda(\mu(G))\leq s$ for each $s\geq |V(\mu(G))|$ and $\lambda(G\times \hat{K}_2)\leq s$ for each $s\geq |V(G\times \hat{K}_2)|$. Using these results, we easily determine $L(2,1)$-labelling numbers and $L'(2,1)$-labelling numbers of several classes of graphs..