On a K_2,3 in non-Hamiltonian graphs

06/25/2019

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by   Heping Jiang, et al.

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In a reduced graph [1], a I cycle-set is the set of cycles only connected by interior points. |I| is the number of I cycle-sets in a given graph. We use a norm graph to denote a reduced graph of |I|=1. An operation denotes a procedure to delete removable cycles. We say an operation is rational means that to delete removable cycles if there have no solutions of Grinberg's Equation of the graph, to delete co-solution cycles if there have solutions of Grinberg's Equation of the graph, or to delete removable cycles if there is a K (a boundary point of order 4) in the graph. g is a subgraph derived by rational operations from a graph G such that there have no removable cycles. In this paper, we present a theorem that a graph G is non-Hamiltonian, if and only if, g and K_2,3 are homeomorphic.