DMGT

Discussiones Mathematicae Graph Theory 27(1) (2007) 179-191
DOI: https://doi.org/10.7151/dmgt.1354

CYCLES THROUGH SPECIFIED VERTICES IN TRIANGLE-FREE GRAPHS

Daniel Paulusma

Department of Computer Science, Durham University
Science Laboratories, South Road, Durham DH1 3LE, England
e-mail: daniel.paulusma@durham.ac.uk

Kiyoshi Yoshimoto

Department of Mathematics
College of Science and Technology
Nihon University, Tokyo 101-8308, Japan
e-mail: yosimoto@math.cst.nihon-u.ac.jp

Abstract

Let G be a triangle-free graph with δ(G) ≥ 2 and σ₄(G) ≥ |V(G)|+2. Let S ⊂ V(G) consist of less than σ₄/4+ 1 vertices. We prove the following. If all vertices of S have degree at least three, then there exists a cycle C containing S. Both the upper bound on |S| and the lower bound on σ₄ are best possible.

Keywords: cycle, path, triangle-free graph.

2000 Mathematics Subject Classification: 05C38, 05C45.

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Received 8 March 2006
Revised 31 October 2006