Abstract
A graph G is called induced matching extendable (shortly, IM-extendable) if every induced matching of G is included in a perfect matching of G. A graph G is called strongly IM-extendable if every spanning supergraph of G is IM-extendable. The k-th power of a graph G, denoted by Gk, is the graph with vertex set V(G) in which two vertices are adjacent if and only if the distance between them in G is at most k.
We obtain the following two results which give positive answers to two conjectures of Yuan.
Result 1. If a connected graph G with |V(G)| even is locally connected, then G2 is strongly IM-extendable.
Result 2. If G is a 2-connected graph with |V(G)| even, then G3 is strongly IM-extendable.
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References
Bondy, J.A., Murty, U.S.R.: Graph Theory with Applications, London: Macmillan Press Ltd 1976
Cameron, K.: Induced matchings. Discrete Appl. Math. 24, 97–102 (1989)
Faudree, R.T., Gyarfas, A., Schelp, R.M., Tuza, Z.: Induced matchings in bipartite graphs. Disrete Math. 78, 83–87 (1989)
Liu, Y., Yuan, J.J., Wang, S.Y.: Degree conditions of IM-extendable graphs. Appl. Math.–JCU. 15B(1), 1–6 (2000)
Lovász, L., Plummer, M.D.: Matching Theory. B. V. North Holland: Elsevier Science Publishers 1985
Wang, Q., Yuan, J.J.: Maximal IM-unextendable graphs, Discrete Math. 240, 295–298 (2001)
Yang, F., Yuan, J.J.: Induced matching extendable graphs without K4 minor, in: Advances in Operations Research and Systems Engineering (Edited by J.F. Gu, G.H. Fan, S.Y. Wang and B. Wei), 1998, 142–145
Yang, F., Yuan, J.J.: NP-completeness of induced matching problem and co-NP-completeness of induced matching extendable problem, OR Transactions. 4(1), 76–80 (2000)
Yuan, J.J.: Induced matching extendable graphs, J. Graph Theory 28, 203–213 (1998)
Yuan, J.J.: Three conjectures on the IM-extendability of the power of a graph, Privite communication
Yuan, J.J.: Induced matching extendable graphs - A survey, Invited Report of the 6th Conference on Operation Research of China, Changsha, China, 2000
Yuan, J.J.: Independent-set-deletable factor-critical graph powers, In submission
Yuan, J.J. and Wang, Q.: Induced matching extendability of G3, Graph Theory Notes of New York, XLIII, 16–19 (2002)
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Research Supported by NSFC Fund 10371102.
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Qian, J. Induced Matching Extendable Graph Powers. Graphs and Combinatorics 22, 391–398 (2006). https://doi.org/10.1007/s00373-006-0673-0
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DOI: https://doi.org/10.1007/s00373-006-0673-0