Abstract
Three theorems are proposed in this paper. The first theorem is that a connected undirected graph G is an Euler graph if and only if G can be expressed as the union of two circles without overlapped sides. Namely, equation satisfies. The second theorem is that a connected simple undirected graph is a Hamilton graph if and only if G contains a sub-graph generated by union of circles of sub-graphs derived from two endpoints of common side. Namely, the equation satisfies (meaning of symbols in the equations see main body of this paper). The third theorem is that a connected simple undirected graph is a Hamilton graph if and only if the loop sum of two circles, and, of sub-graphs derived from two endpoints of common side in graph G is a sub-graphs of loop graph Cn.
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