Most Graphs are Edge-Cordial
نویسندگان
چکیده
We extend the de nition of edge-cordial graphs due to Ng and Lee for graphs on 4k, 4k+1, and 4k+3 vertices to include graphs on 4k+2 vertices, and show that, in fact, all graphs without isolated vertices are edge-cordial. Ng and Lee conjectured that all trees on 4k, 4k + 1, or 4k + 3 vertices are edge-cordial. Intuitively speaking, a graph G is said to be edge-cordial if its edges can be labelled either 0 or 1 so that half of the edges are labelled 0; half are labelled 1; and half of the vertices meet an even number of edges labelled 1, while the other half meet an odd number of edges labelled 1. This de nition is due to Ng and Lee, see [2], and is the dual concept to cordial graphs, rst introduced by Cahit in [1]. Ng and Lee showed that if the number of vertices in any graph is congruent to 2 (modulo 4), then the graph cannot be edge-cordial. We extend their de nition of edge-cordial to graphs on 4k+ 2 vertices. De nition 1 Suppose we have a simple graph G = (V;E). Let f : E ! f0; 1g be an edge-labelling of G. De ne a vertex-labelling f : V ! f0; 1g by f (v) = X (u;v)2E f(u; x) (mod 2) Let Ei denote the set of edges labelled i, and Vi be the set of vertices labelled i, for i = 0; 1. Then G is said to be edge-cordial if and only if 1. (Ng and Lee) j V j 0; 1; 3 (mod 4) and there exists an edge-labelling f such that j (j E0 j j E1 j) j 1 and j (j V0 j j V1 j) j 1; or
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