Partition the vertices of a graph into induced matchings

  • Authors:

  • Jinjiang Yuan;Qin Wang

  • Affiliations:

  • Department of Mathematics, Zhengzhou University, Zhengzhou, Henan 450052, People's Republic of China;Department of Mathematics, Zhengzhou University, Zhengzhou, Henan 450052, People's Republic of China and Department of Mathematics, City University of Hong Kong, Hong Kong, People's Republic of Ch ...

  • Venue:
  • Discrete Mathematics
  • Year:
  • 2003

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Abstract

The induced matching partition number of a graph G, denoted by imp(G), is the minimum integer k such that V(G) has a k-partition (V1, V2,..., Vk) such that, for each i, 1 ≤ i ≤ k, G[Vi], the subgraph of G induced by Vi, is a 1-regular graph. This is different from the strong chromatic index--the minimum size of a partition of the edges of graph into induced matchings. It is easy to show, as we do in this paper, that, if G is a graph which has a perfect matching, then imp(G) ≤ 2Δ (G) - 1, where Δ(G) is the maximum degree of a vertex of G. We further show in this paper that, when G is connected, imp(G) = 2 Δ(G) - 1 if and only if G is isomorphic to either K2 or C4k+2 or the Petersen graph, where. Cn is the cycle of length n.