DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

https://doi.org/10.7151/dmgt

Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory 24(2) (2004) 249-262
DOI: 10.7151/dmgt.1229

VERTEX-DISJOINT COPIES OF K4-

Ken-ichi Kawarabayashi

Department of Mathematics
Faculty of Science and Technology, Keio University
3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan

e-mail: k_keniti@comb.math.keio.ac.jp

Abstract

Let G be a graph of order n. Let Kl- be the graph obtained from Kl by removing one edge.

In this paper, we propose the following conjecture:

Let G be a graph of order n ≥ lk with δ (G) ≥ (n-k+1)[(l-3)/(l-2)]+k-1. Then G has k vertex-disjoint Kl-.

This conjecture is motivated by Hajnal and Szemerédi's [6] famous theorem.

In this paper, we verify this conjecture for l=4.

Keywords: extremal graph theory, vertex disjoint copy, minimum degree.

2000 Mathematics Subject Classification: 05C70, 05C38.

References

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[3] Y. Egawa and K. Ota, Vertex-Disjoint K1,3 in graphs, Discrete Math. 197/198 (1999), 225-246.
[4] Y. Egawa and K. Ota, Vertex-disjoint paths in graphs, Ars Combinatoria 61 (2001) 23-31.
[5] Y. Egawa and K. Ota, K1,3-factors in graphs, preprint.
[6] A. Hajnal and E. Szemerédi, Proof of a conjecture of P. Erdős, Colloq. Math. Soc. János Bolyai 4 (1970) 601-623.
[7] K. Kawarabayashi, K4-factor in a graph, J. Graph Theory 39 (2002) 111-128, doi: 10.1002/jgt.10007.
[8] K. Kawarabayashi, F-factor and vertex disjoint F in a graph, Ars Combinatoria 62 (2002) 183-187.
[9] J. Komlós, Tiling Túran theorems, Combinatorica 20 (2000) 203-218.

Received 31 August 2002
Revised 6 February 2004