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A note on two circumference generalizations of Chvátal’s Hamiltonicity condition. (English) Zbl 1006.05036
Generalizing the well-known Chvátal’s Hamiltonicity condition B. Bollobás in his classical book on extremal graph theory asked: Let \(G\) be a \(2\)-connected graph of order \(n\) with vertex degrees \(d_1 \leq d_2 \leq \dots \leq d_n\). Suppose \(3 \leq c \leq n\) and \(d_i \leq i < c/2\) implies \(d_{n-i} \geq c-i\). Does it follow that for the circumference \(c(G)\) of \(G\) (the length of its longest cycle) \(c(G) \geq c\) holds? For \(c=3\) and \(c=4\) this follows directly from the \(2\)-connectivity of \(G\), however Häggkvist found counterexamples for any \(c \geq 7\). In the paper the author proves that for \(c=5\) the answer is affirmative while for \(c=6\) counterexamples are presented. An interesting generalization of Chvátal’s condition for any \(c\) is formulated and proved.
MSC:
05C38 Paths and cycles
05C45 Eulerian and Hamiltonian graphs
05C07 Vertex degrees
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References:
[1] BEDROSSIAN P., CHEN G., SCHELP R. H.: A generalization of Fan’s condition for hamiltonicity, pancyclicity, and hamiltonian connectedness. Discrete Math. 115 (1993), 39-50. · Zbl 0773.05075
[2] BOLLOBÁS B.: Extremal Graph Theory. Acad. Press, London-New York-San Francisco, 1978. · Zbl 1099.05044
[3] BOLLOBÁS B., BRIGHTWELL G.: Cycles through specified vertices. Combinatorica 13 (1993), 147 -155. · Zbl 0780.05033
[4] BONDY J. A.: Large cycles in graphs. Discrete Math. 1 (1971), 121-132. · Zbl 0224.05120
[5] BONDY J. A., LOVÁSZ L.: Cycles through specified vertices of a graph. Combinatorica 1 (1981). 117-140. · Zbl 0492.05049
[6] HÄGGKVIST R.: · Zbl 1218.05129
[7] CHVÁTAL V.: On hamilton’s ideals. J. Combin. Theory Ser. B 12 (1972), 163-168. · Zbl 0213.50803
[8] SHI R.: 2-Neighborhoods and hamiltonian conditions. J. Graph Theory 16 (1992), 267-271. · Zbl 0761.05066
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