Long cycles in bipartite graphs

Long cycles in unbalanced bipartite graphs

Let G[X,Y ] be a 2-connected bipartite graph with |X| ≥ |Y |. For S ⊆ V (G), we define δ(S;G) := min{dG(v) : v ∈ S}. We define σ1,1(G) := min{dG(x) + dG(y) : x ∈ X, y ∈ Y, xy / ∈ E(G)} and σ2(X) := min{dG(x) + dG(y) : x, y ∈ X,x 6= y}. We denote by c(G) the length of a longest cycle in G. Jackson [J. Combin. Theory Ser. B 38 (1985), 118–131] proved that c(G) ≥ min{2δ(X;G) + 2δ(Y ;G)− 2, 4δ(X;G)...

Edge condition for long cycles in bipartite graphs

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Balanced cycles and holes in bipartite graphs

Bruce Reed asks the following question: Can we determine whether a bipartite graph contains a chordless cycle whose length is a multiple of 4? We show that the two following more general questions are equivalent and we provide an answer. Given a bipartite graph G where each edge is assigned a weight + 1 or I, l determine whether G contains a cycle whose weight is a multiple of 4, l determine wh...

Enumerating and Counting Cycles in Bipartite Graphs

Disjoint hamiltonian cycles in bipartite graphs

Let G = (X, Y ) be a bipartite graph and define σ 2(G) = min{d(x) + d(y) : xy / ∈ E(G), x ∈ X, y ∈ Y }. Moon and Moser [5] showed that if G is a bipartite graph on 2n vertices such that σ 2(G) ≥ n + 1 then G is hamiltonian, sharpening a classical result of Ore [6] for bipartite graphs. Here we prove that if G is a bipartite graph on 2n vertices such that σ 2(G) ≥ n+ 2k− 1 then G contains k edge...