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A degree condition for the circumference of a graph

✍ Scribed by Nathaniel Dean; Pierre Fraisse

Publisher: John Wiley and Sons
Year: 1989
Tongue: English
Weight: 198 KB
Volume: 13
Category: Article
ISSN: 0364-9024
DOI: 10.1002/jgt.3190130308

No coin nor oath required. For personal study only.

✦ Synopsis

We present a new condition on the degree sums of a graph that implies the existence of a long cycle. Let c(G) denote the length of a longest cycle in the graph G and let rn be any positive integer. Suppose G is a 2-connected graph with vertices x,, . . . , x, and edge set E that satisfies the property that, for any two integers j and k with j < k, xixk 4 E, d(x,) I j and d(x,) 5 k -1, we have ( 1

either rn 2 n or d(x,) + d(xk) 2 min(k + 1, rn). Then c ( G ) L min(rn, n). This result unifies previous results of J. C. Bermond and M. Las Vergnas, respectively.

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