# Symmetric Hamilton Cycle Decompositions of Complete Graphs Plus a 1-Factor

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Assume that n and Î´ are positive integers with 2 â‰¤ Î´ < n. Let h(n, Î´) be the minimum number of edges required to guarantee an n-vertex graph with minimum degree Î´(G) â‰¥ Î´ to be hamiltonian, i.e., any n-vertex graph G with Î´(G) â‰¥ Î´ is hamiltonian if |E(G)|...

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For n = 4, the complete n-vertex multidigraph with arc multiplicity ? is proved to have a decomposition into directed paths of arbitrarily prescribed lengths = n-1 and different from n-2, unless n = 5, ? = 1, and all lengths are to be n-1 = 4. For ? = 1, a more general decomposition exists;...

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We prove that a complete bipartite graph can be decomposed into cycles of arbitrary specified lengths provided that the obvious necessary conditions are satisfied, the length of each cycle is at most the size of the smallest part, and the longest cycle is at most three times as long as the...

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Let F be a certain graph, the graph F-Path denoted by â„™ d+1(F) path of length d with d + 1 vertices (i.e. Every edge of this path is one-to-one corresponding to an isomorphic to the graph F). In the same manner, we define the graph F-Cycle as â„‚d(F) cycle on d vertices. In this...

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For n â‰¥ 4, the complete n-vertex multidigraph with arc multiplicity Î» is proved to have a decomposition into directed paths of arbitrarily prescribed lengths â‰¤ n--1 and different from n--2, unless n = 5, Î» = 1, and all lengths are to be n--1 = 4. For Î»= 1, a more general...