# DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

# Discussiones Mathematicae Graph Theory

Journal Impact Factor (JIF 2023): 0.5

5-year Journal Impact Factor (2023): 0.6

CiteScore (2023): 2.2

SNIP (2023): 0.681

# Discussiones Mathematicae Graph Theory

## 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