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Extending cycles in bipartite graphs. (English) Zbl 0674.05044
Let G(X,Y,E) be a balanced bipartite graph of order 2n. We introduce the following definitions. A cycle C in G is extendable if there exists a cycle C’ in G such that V(C)$\subseteq V(C')$ and $\vert V(C')\vert =\vert V(C)\vert +2$. G is bi-cycle extendable if G has at least one cycle and every nonhamiltonian cycle in G is extendable. G has a bi- pancyclic ordering if the vertices of X and Y can be labelled $x\sb 1,x\sb 2,...,x\sb n$ and $y\sb 1,y\sb 2,...,y\sb n$, respectively, so that $C\sb{2k}\subseteq <x\sb 1,...,x\sb k,y\sb 1,...,y\sb k>,$ for $2\le k\le n.$ Let $$ {\bar \sigma}(G)=\min \{d(x)+d(y):\quad x\in X,\quad y\in Y\quad and\quad xy\not\in E(G)\}. $$ It is shown that if ${\bar \sigma}$(G)$\ge n+1$ and C is a 2k-cycle in G then C is extendable unless $<V(C)>\cong K\sb{k,k}$. As consequences of the proof of this result, we deduce that if either ${\bar \sigma}(G)\ge (7n+1)/6$ or $\delta (G)\ge (n+1)/2$ then, in each case with one exceptional graph, G is bi-cycle extendable. It is also shown that if $\ell$ is an integer such that $n\ge 2\ell \ge 2,$ $\delta (G)\ge \ell$ and $\vert E(G)\vert \ge n\sp 2-\ell n+\ell\sp 2$ then every cycle of length at least $\ell$ in G is extendable unless $G\cong K\sb{n,n}-E(K\sb{\ell,n-\ell}).$ As a corollary, we deduce that such graph G has a bi-pancyclic ordering unless $G\cong K\sb{n,n}-E(K\sb{\ell,n-\ell}).$ A number of preliminary results are required, among which is the determination of the maximum size of a balanced bipartite graph of specified order, minimum degree and edge independence number.
Reviewer: G.R.T.Hendry

MSC:
05C38Paths; cycles
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References:
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