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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
##### Keywords:
circumference; Hamiltonicity condition; degree sequence
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##### References:
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