Discussiones Mathematicae Graph Theory 32(2) (2012) 331-339
doi: 10.7151/dmgt.1604

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On Ramsey (K1, 2, Kn)-minimal Graphs

Mariusz Hałuszczak

Faculty of Mathematics, Computer Science and Econometrics
University of Zielona Góra
Z. Szafrana 4a, Zielona Góra, Poland

Abstract

Let F be a graph and let G, H denote nonempty families of graphs. We write F → (G, H) if in any 2-coloring of edges of F with red and blue, there is a red subgraph isomorphic to some graph from G or a blue subgraph isomorphic to some graph from H. The graph F without isolated vertices is said to be a (G, H)-minimal graph if F → (G, H) and F −e not → (G, H) for every e ∈ E(F).

We present a technique which allows to generate infinite family of (G, H)-minimal graphs if we know some special graphs. In particular, we show how to receive infinite family of (K1, 2, Kn)-minimal graphs, for every n ≥ 3.

Keywords: Ramsey minimal graph, edge coloring, 1-factor, complete graph

2010 Mathematics Subject Classification: 05C55, 05C70, 05C76, 05D10.

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Received 13 October 2010
Revised 20 June 2011
Accepted 20 June 2011