Extremal bipartite graphs with a unique k-factor

• Autorzy: Arne Hoffmann , Elżbieta Sidorowicz , Lutz Volkmann
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

unique k-factor; bipartite graphs; extremal graphs

Kategorie tematyczne

• 05C70: Factorization, matching, partitioning, covering and packing
• 05C35: Extremal problems

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2006
• Tom: 26
• Numer: 2
• Strony: 181-192

Daty

wydano

2006

otrzymano

2002-03-14

poprawiono

2005-12-15

Twórcy

autor

Arne Hoffmann

• Watson Wyatt Deutschland GmbH, 80339 Munich, Germany

autor

Elżbieta Sidorowicz

• Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Szafrana 4a, 65-516 Zielona Góra, Poland

autor

Lutz Volkmann

• Lehrstuhl II für Mathematik, RWTH-Aachen, 52056 Aachen, Germany

Bibliografia

• [1] G. Chartrand and L. Lesniak, Graphs and Digraphs 3rd edition (Chapman and Hall, London 1996).
• [2] G.R.T. Hendry, Maximum graphs with a unique k-factor, J. Combin. Theory (B) 37 (1984) 53-63, doi: 10.1016/0095-8956(84)90044-3.
• [3] A. Hoffmann and L. Volkmann, On unique k-factors and unique [1,k] -factors in graphs, Discrete Math. 278 (2004) 127-138, doi: 10.1016/S0012-365X(03)00248-6.
• [4] P. Johann, On the structure of graphs with a unique k-factor, J. Graph Theory 35 (2000) 227-243, doi: 10.1002/1097-0118(200012)35:4<227::AID-JGT1>3.0.CO;2-D
• [5] D. König, Über Graphen und ihre Anwendung auf Determinantentheorie und Mengenlehre, Math. Ann. 77 (1916) 453-465, doi: 10.1007/BF01456961.
• [6] J. Sheehan, Graphs with exactly one hamiltonian circuit, J. Graph Theory 1 (1977) 37-43, doi: 10.1002/jgt.3190010110.
• [7] L. Volkmann, The maximum size of graphs with a unique k-factor, Combinatorica 24 (2004) 531-540, doi: 10.1007/s00493-004-0032-9.

Identyfikatory

DOI

10.7151/dmgt.1311