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adjacent algorithm bipartite graph c-trees chordal graphs circuit clique color Combinatorial complete graph component Computer conjecture connected graph consider construction Corollary cycle decomposition defined deleting denote diameter digraph disjoint union edge-connectivity edge-disconnecting set edge-disjoint edges of G elements embedding eulerian Eulerian Graphs example exists fc-path irregular Figure finite graph contains graph G graph of order Graph Theory grid graph Hamiltonian Hamiltonian path Harary Hence independent set induced subgraph integer irregular graphs isomorphic k-tree knot labels least Lemma Let G linear lower bound matching Math Mathematics matroid maximum number minimal minimum network reliability node non-empty subset nontrivial number of edges number of vertices obtained optimal paper partition path of length permutation graphs planar graph polynomial positive integer problem proof of Theorem Proposition result series class Shellsort spanning trees Suppose symmetric group symmetry topological upper bounds vectors vertex set vertices of degree vertices of G