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Let I with |I| = k be a matching of a graph G (briefly, I is called a k-matching). If I is not a proper subset of any other matching of G, then I is a maximal k-matching and m(gk,G) is used to denote the number of maximal k-matchings of G. Let gk be a k-matching of G, if there exists a subset {e1, e2, …, ei} of E(G) \ gk, i ≥ 1, such that (1) for any j ɛ {1, 2, …, i}, gk + {ej} is a (k + 1)-matching...
The total chromatic number χT(G) of graph G is the least number of colors assigned to VE(G) such that no adjacent or incident elements receive the same color. Given graphs G1, G2, the join of G1 and G2, denoted by G1 ∨ G2, is a graph G, V(G) = V(G1) ∪ V(G2) and E(G) = E(G1) ∪ E(G2) ∪ {uv | u ε V(G1), v ε V(G2)}. In this paper, it's proved that if v(G) = v(H), both Gc and Hc contain perfect matching...
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