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 26(2) (2006) 181-192
DOI: 10.7151/dmgt.1311

EXTREMAL BIPARTITE GRAPHS WITH A UNIQUE k-FACTOR

Arne Hoffmann

Watson Wyatt Deutschland GmbH
80339 Munich, Germany
e-mail: arne.hoffmann@eu.watsonwyatt.com

Elżbieta Sidorowicz

Faculty of Mathematics, Computer Science and Econometrics
University of Zielona Góra
Szafrana 4a, 65-516 Zielona Góra, Poland
e-mail: e.sidorowicz@wmie.uz.zgora.pl

Lutz Volkmann

Lehrstuhl II für Mathematik, RWTH-Aachen
52056 Aachen, Germany
e-mail: volkm@math2.rwth-aachen.de

Abstract

Given integers p > k > 0, we consider the following problem of extremal graph theory: How many edges can a bipartite graph of order 2p have, if it contains a unique k-factor? We show that a labeling of the vertices in each part exists, such that at each vertex the indices of its neighbours in the factor are either all greater or all smaller than those of its neighbours in the graph without the factor. This enables us to prove that every bipartite graph with a unique k-factor and maximal size has exactly 2k vertices of degree k and 2k vertices of degree [(|V(G)|)/2]. As our main result we show that for k ≥ 1, p ≡ t mod k, 0 ≤ t < k, a bipartite graph G of order 2p with a unique k-factor meets 2|E(G)| ≤ p(p+k)−t(k−t). Furthermore, we present the structure of extremal graphs.

Keywords: unique k-factor, bipartite graphs, extremal graphs.

Mathematics Subject Classification: Primary 05C70; Secondary 05C35.

References

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Received 14 March 2002
Revised 15 December 2005