## 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 $$| V(C')| =| V(C)| +2$$. 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 $$x_ 1,x_ 2,...,x_ n$$ and $$y_ 1,y_ 2,...,y_ n$$, respectively, so that $$C_{2k}\subseteq <x_ 1,...,x_ k,y_ 1,...,y_ k>,$$ for $$2\leq k\leq n.$$ 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)$$\geq n+1$$ and C is a 2k-cycle in G then C is extendable unless $$<V(C)>\cong K_{k,k}$$. As consequences of the proof of this result, we deduce that if either $${\bar \sigma}(G)\geq (7n+1)/6$$ or $$\delta (G)\geq (n+1)/2$$ 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 $$n\geq 2\ell \geq 2,$$ $$\delta (G)\geq \ell$$ and $$| E(G)| \geq n^ 2-\ell n+\ell^ 2$$ then every cycle of length at least $$\ell$$ in G is extendable unless $$G\cong K_{n,n}-E(K_{\ell,n-\ell}).$$ As a corollary, we deduce that such graph G has a bi-pancyclic ordering unless $$G\cong K_{n,n}-E(K_{\ell,n-\ell}).$$
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:

 05C38 Paths and cycles

### Keywords:

Hamiltonian; bipartite graph; cycle
Full Text:

### References:

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