Nordhaus–Gaddum type result for the matching number of a graph

Abstract

For a graph G, $\alpha '(G)$ is the matching number of G. Let $k\ge 2$ be an integer, $K_{n}$ be the complete graph of order n. Assume that $G_{1}, G_{2}, \ldots , G_{k}$ is a k-decomposition of $K_{n}$. In this paper, we show that (1)

$$\begin{aligned} \left\lfloor \frac{n}{2}\right\rfloor \le \sum _{i=1}^{k} \alpha '(G_{i})\le k\left\lfloor \frac{n}{2}\right\rfloor . \end{aligned}$$

(2) If each $G_{i}$ is non-empty for $i = 1, \ldots , k$, then for $n\ge 6k$,

$$\begin{aligned} \sum _{i=1}^{k} \alpha '(G_{i})\ge \left\lfloor \frac{n+k-1}{2}\right\rfloor . \end{aligned}$$

(3) If $G_{i}$ has no isolated vertices for $i = 1, \ldots , k$, then for $n\ge 8k$,

$$\begin{aligned} \sum _{i=1}^{k} \alpha '(G_{i})\ge \left\lfloor \frac{n}{2}\right\rfloor +k. \end{aligned}$$

The bounds in (1), (2) and (3) are sharp. (4) When $k= 2$, we characterize all the extremal graphs which attain the lower bounds in (1), (2) and (3), respectively.