Partitioning a Graph into Two Square-Cycles

Genghua Fan; Henry Kierstead

doi:10.1002/(SICI)1097-0118(199611)23:3<241::AID-JGT4>3.0.CO;2-S

Partitioning a Graph into Two Square-Cycles

Genghua Fan, Henry Kierstead

Mathematical and Statistical Sciences, School of (SoMSS)

Research output: Contribution to journal › Article › peer-review

9 Scopus citations

Abstract

A square-cycle is the graph obtained from a cycle by joining every pair of vertices of distance two in the cycle. The length of a square-cycle is the number of vertices in it. Let G be a graph on n vertices with minimum degree at least 2/3n and let c be the maximum length of a square-cycle in G. Pósa and Seymour conjectured that c = n. In this paper, it is proved that either c = n or 1/2n ≤ c ≤ 2/3n. As an application of this result, it is shown that G has two vertex-disjoint square-cycles C₁ and C₂ such that V(G) = V(C₁) ∪ V(C₂).

Original language	English (US)
Pages (from-to)	241-256
Number of pages	16
Journal	Journal of Graph Theory
Volume	23
Issue number	3
DOIs	https://doi.org/10.1002/(SICI)1097-0118(199611)23:3<241::AID-JGT4>3.0.CO;2-S
State	Published - Nov 1996

ASJC Scopus subject areas

Geometry and Topology

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10.1002/(SICI)1097-0118(199611)23:3<241::AID-JGT4>3.0.CO;2-S

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title = "Partitioning a Graph into Two Square-Cycles",

abstract = "A square-cycle is the graph obtained from a cycle by joining every pair of vertices of distance two in the cycle. The length of a square-cycle is the number of vertices in it. Let G be a graph on n vertices with minimum degree at least 2/3n and let c be the maximum length of a square-cycle in G. P{\'o}sa and Seymour conjectured that c = n. In this paper, it is proved that either c = n or 1/2n ≤ c ≤ 2/3n. As an application of this result, it is shown that G has two vertex-disjoint square-cycles C1 and C2 such that V(G) = V(C1) ∪ V(C2).",

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AB - A square-cycle is the graph obtained from a cycle by joining every pair of vertices of distance two in the cycle. The length of a square-cycle is the number of vertices in it. Let G be a graph on n vertices with minimum degree at least 2/3n and let c be the maximum length of a square-cycle in G. Pósa and Seymour conjectured that c = n. In this paper, it is proved that either c = n or 1/2n ≤ c ≤ 2/3n. As an application of this result, it is shown that G has two vertex-disjoint square-cycles C1 and C2 such that V(G) = V(C1) ∪ V(C2).

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Partitioning a Graph into Two Square-Cycles

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