# Relative length of long paths and cycles in graphs with large degree sums

Hikoe Enomoto, Jan van den Heuvel, Atsushi Kaneko, Akira Saito

20 Citations (Scopus)

### Abstract

For a graph G, p(G) denotes the order of a longest path in G and c(G) the order of a longest cycle. We show that if G is a connected graph n ≥ 3 vertices such that d(u) + d(v) + d(w) n for all triples u, v, w of independent vertices, then G satisfies c(G) ≥ p(G) - 1, or G is in one of six families of exceptional graphs. This generalizes results of Bondy and of Bauer, Morgana, Schmeichel, and Veldman.
Original language English 213-225 Journal of graph theory 201 2 https://doi.org/10.1002/jgt.3190200210 Published - 1995

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### Cite this

Enomoto, Hikoe ; van den Heuvel, Jan ; Kaneko, Atsushi ; Saito, Akira. / Relative length of long paths and cycles in graphs with large degree sums. In: Journal of graph theory. 1995 ; Vol. 201, No. 2. pp. 213-225.
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Enomoto, H, van den Heuvel, J, Kaneko, A & Saito, A 1995, 'Relative length of long paths and cycles in graphs with large degree sums' Journal of graph theory, vol. 201, no. 2, pp. 213-225. https://doi.org/10.1002/jgt.3190200210

Relative length of long paths and cycles in graphs with large degree sums. / Enomoto, Hikoe; van den Heuvel, Jan; Kaneko, Atsushi; Saito, Akira.

In: Journal of graph theory, Vol. 201, No. 2, 1995, p. 213-225.

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T1 - Relative length of long paths and cycles in graphs with large degree sums

AU - Enomoto, Hikoe

AU - van den Heuvel, Jan

AU - Kaneko, Atsushi

AU - Saito, Akira

PY - 1995

Y1 - 1995

N2 - For a graph G, p(G) denotes the order of a longest path in G and c(G) the order of a longest cycle. We show that if G is a connected graph n ≥ 3 vertices such that d(u) + d(v) + d(w) n for all triples u, v, w of independent vertices, then G satisfies c(G) ≥ p(G) - 1, or G is in one of six families of exceptional graphs. This generalizes results of Bondy and of Bauer, Morgana, Schmeichel, and Veldman.

AB - For a graph G, p(G) denotes the order of a longest path in G and c(G) the order of a longest cycle. We show that if G is a connected graph n ≥ 3 vertices such that d(u) + d(v) + d(w) n for all triples u, v, w of independent vertices, then G satisfies c(G) ≥ p(G) - 1, or G is in one of six families of exceptional graphs. This generalizes results of Bondy and of Bauer, Morgana, Schmeichel, and Veldman.

KW - IR-71177

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DO - 10.1002/jgt.3190200210

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EP - 225

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