INDEPENDENT CYCLES AND PATHS IN BIPARTITE BALANCED GRAPHS

Beata Orchel  and  A. Paweł Wojda

Faculty of Applied Mathematics
AGH University of Science and Technology
Al. Mickiewicza 30, 30-059 Kraków, Poland
e-mail: orchel@uci.agh.edu.pl
e-mail: wojda@uci.agh.edu.pl

Abstract

Bipartite graphs G = (L,R;E) and H = (L′,R′;E′) are bi-placeabe if there is a bijection f:L∪R→ L′∪R′ such that f(L) = L′ and f(u)f(v) ∉ E′ for every edge uv ∈ E. We prove that if G and H are two bipartite balanced graphs of order |G| = |H| = 2p ≥ 4 such that the sizes of G and H satisfy ||G|| ≤ 2p−3 and ||H|| ≤ 2p−2, and the maximum degree of H is at most 2, then G and H are bi-placeable, unless G and H is one of easily recognizable couples of graphs.

This result implies easily that for integers p and k1,k2,...,kl such that ki ≥ 2 for i = 1,... ,l and k1+... + kl ≤ p−1 every bipartite balanced graph G of order 2p and size at least p2−2p+3 contains mutually vertex disjoint cycles C2k1,... ,C2kl, unless G = K3,3−3K1,1.

Keywords: bipartite graphs, bi-placing, path, cycle.

2000 Mathematics Subject Classification: 05C38, 05C35.

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