Vertex-Fault Tolerant Complete Matching in Bipartite graphs: the Biregular Case
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Given a family H of graphs and a positive integer k, a graph G is called vertex k-fault-tolerant with respect to H, denoted by k-FT(H), if G-S contains some H∈H as a subgraph, for every S⊂ V(G) with |S|≤ k. Vertex-fault-tolerance has been introduced by Hayes [A graph model for fault-tolerant computing systems, IEEE Transactions on Computers, C-25 (1976), pp. 875-884.], and has been studied in view of potential applications in the design of interconnection networks operating correctly in the presence of faults. We define the Fault-Tolerant Complete Matching (FTCM) Problem in bipartite graphs of order (n,m): to design a bipartite G=(U,V;E), with |U|=n, |V|=m, n>m>1, that has a FTCM, and the tuple (Δ_U, Δ_V), where Δ_U and Δ_V are the maximum degree in U and V, respectively, is lexicographically minimum. G has a FTCM if deleting at most n-m vertices from U creates G' that has a complete matching, i.e., a matching of size m. We show that if m(n-m+1)/n is integer, solutions of the FTCM Problem can be found among (a,b)-regular bipartite graphs of order (n,m), with a=m(n-m+1)/n, and b=n-m+1. If a=m-1 then all (a,b)-regular bipartite graphs of order (n,m) have a FTCM, and for a<m-1, it is not the case. We characterize the values of n, m, a, and b that admit an (a,b)-regular bipartite graph of order (n,m), with b=n-m+1, and give a simple construction that creates such a graph with a FTCM whenever possible. Our techniques are based on Hall's marriage theorem, elementary number theory, linear Diophantine equations, properties of integer functions and congruences, and equations involving them.
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