# Search Results

## Abstract

The Cartesian product of n cycles is a 2n-regular, 2n-connected and bi- pancyclic graph. Let G be the Cartesian product of n even cycles and let 2n = n_{1}+ n_{2}+ ・ ・ ・ + n_{k}with k ≥ 2 and n_{i}≥ 2 for each i. We prove that if k = 2, then G can be decomposed into two spanning subgraphs G_{1}and G_{2}such that each G_{i}is n_{i}-regular, n_{i}-connected, and bipancyclic or nearly bipancyclic. For k > 2, we establish that if all n_{i}in the partition of 2n are even, then G can be decomposed into k spanning subgraphs G_{1},G_{2}, . . . ,Gk such that each G_{i}is n_{i}-regular and n_{i}-connected. These results are analo- gous to the corresponding results for hypercubes.

## Abstract

Slater introduced the point-addition operation on graphs to characterize 4-connected graphs. The Г-extension operation on binary matroids is a generalization of the point-addition operation. In general, under the Г-extension operation the properties like graphicness and cographicness of matroids are not preserved. In this paper, we obtain forbidden minor characterizations for binary matroids whose Г-extension matroids are graphic (respectively, cographic).

## Abstract

The conditional *h*-vertex (*h*-edge) connectivity of a connected graph *H* of minimum degree *k > h* is the size of a smallest vertex (edge) set *F* of *H* such that *H* − *F* is a disconnected graph of minimum degree at least *h.* Let *G* be the Cartesian product of *r* ≥ 1 cycles, each of length at least four and let *h* be an integer such that 0 ≤ *h* ≤ 2*r* − 2. In this paper, we determine the conditional *h*-vertex-connectivity and the conditional *h*-edge-connectivity of the graph *G.* We prove that both these connectivities are equal to (2*r*−*h*)*a ^{r}_{h}*, where

*a*is the number of vertices of a smallest

^{r}_{h}*h*-regular subgraph of

*G.*