Abstract
We denote by G[X, Y] a bipartite graph G with partite sets X and Y. Let d G (v) be the degree of a vertex v in a graph G. For G[X, Y] and \({S \subseteq V(G),}\) we define \({\sigma_{1,1}(S):=\min\{d_G(x)+d_G(y) : (x,y) \in (X \cap S,Y) \cup (X, Y \cap S), xy \not\in E(G)\}}\) . Amar et al. (Opusc. Math. 29:345–364, 2009) obtained σ 1,1(S) condition for cyclability of balanced bipartite graphs. In this paper, we generalize the result as it includes the case of unbalanced bipartite graphs: if G[X, Y] is a 2-connected bipartite graph with |X| ≥ |Y| and \({S \subseteq V(G)}\) such that σ 1,1(S) ≥ |X| + 1, then either there exists a cycle containing S or \({|S \cap X| > |Y|}\) and there exists a cycle containing Y. This degree sum condition is sharp.
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Okamura, H., Yamashita, T. Degree Sum Conditions for Cyclability in Bipartite Graphs. Graphs and Combinatorics 29, 1077–1085 (2013). https://doi.org/10.1007/s00373-012-1148-0
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DOI: https://doi.org/10.1007/s00373-012-1148-0