Tight Bound for Matching1

Matching is an extensively studied topic. The question we want to study here is the lower bound of a maximum matching in a graph. Earlier researchers studied the problem of the existence of a perfect matching (i.e. a matching of size n/2 with n vertices in a graph). Petersen [7] showed that a bridgeless cubic graph has a perfect matching. König [4] showed that there exists a perfect matching in any k-regular bipartite graph. Tutte [8] characterizes when a graph has a perfect matching. For graphs without a perfect matching the size of a maximum matching is studied. Nishizeki and Baybars [6] showed that any 3-connected planar graph has a matching of size at least (n+ 4)/3 for n ≥ 22. Biedl et al. [1] raised the question whether a bound better than m/(2k − 1) can be obtained for the size of a maximum matching in a graph of m edges and degree k. Feng et al. [2] showed a lower bound of 2m/(3k− 1) (for k ≥ 3) for this problem. Recently Han [3] showed a lower bound of 4m/(5k + 3). Actually Vizing has shown [9] that graphs with degree k can be edge colored with k+ 1 colors. Misra and Gries gave a constructive proof of this [5]. This result will give a matching of size m/(k + 1) and is better than the results shown in [1, 2, 3]. The author forgot about this known result and worked after the results in [1, 2]. After presenting the paper [3] at FAW’2008 Daniel Lokshtanov reminded me of Vizing’s theorem. After examining our techniques used in [3] I informed Daniel Lokshtanov right way that the technique employed in [3] will allow me to outperform the result obtainable from Vizing’s theorem. Here we present our result of a tight bound on matching which outperforms that achievable via Vizing’s theorem.