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Two theorems on Hamiltonian graphs. (English. Russian original) Zbl 0552.05038
Math. Notes 35, 32-35 (1984); translation from Mat. Zametki 35, No. 1, 55-61 (1984).
In the paper two conditions for the hamiltonicity of a graph are presented. The number of vertices of a graph G is denoted by p, the degree of a vertex x in G by \(d_ G(x)\) and the set of vetices adjacent to x in G by \(N_ G(x)\). Further \(\epsilon_ G(x)=\cup_{y\in N_ G(x)}N_ G(y)\), \({\bar \epsilon}_ G(x)=\{y\in \epsilon_ G(x)| d_ G(y)\leq d_ G(x)\}\). It is always supposed \(p\geq 3\). A graph G has the property \(\xi\), if it has no isolated vertices and for any vertex x of G the inequality \(d_ G(x)<(p-1)/2\) implies \(| {\bar \epsilon}_ G(x)| <d_ G(x)\) and the equality \(d_ G(x)=(p-1)/2)\) implies \(| {\bar \epsilon}_ G(x)| \leq d_ G(x).\) The p-closure \(C_ p(G)\) of G is the (unique) graph H whose vertex set is equal to the vertex set of G, whose edge set contains the edge set of G as a subset, in which \(d_ H(x)+d_ H(y)<p\) for any two non-adjacent vertices x, y holds and which has the minimum number of edges from all graphs with these properties. It is proved that if G or \(C_ p(G)\) has the property \(\xi\), then G is Hamiltonian.
Reviewer: B.Zelinka

MSC:
05C45 Eulerian and Hamiltonian graphs
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