Let G be a graph obtained by taking the Cartesian product of finitely many cycles. It is known that G is bipancyclic, that is, G contains cycles of every even length from 4 to |V (G)|. We extend this result for the existence of 3-regular subgraphs in G. We prove that G contains a 3-regular, 2-connected subgraph with l vertices if l = 8 or l = 12 or l is an even integer with 16 ≤ l ≤ |V (G)|. For l ∈ {6, 10, 14}, we give necessary and sufficient conditions for the existence of such subgraphs in G. c
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 ≤ 2r − 2. In this paper, we determine the conditional h-vertex-connectivity and the conditional hedge -connectivity of the graph G. We prove that both these connectivities are equal to (2r −h)a r h , where a r h is the number of vertices of a smallest h-regular subgraph of G.
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 ≤ 2r − 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 (2r − h)a r h , where a r h is the number of vertices of a smallest h-regular subgraph of G.
Let [Formula: see text] be a connected graph with minimum degree at least [Formula: see text] and let [Formula: see text] be an integer such that [Formula: see text] The conditional [Formula: see text]-edge ([Formula: see text]-vertex) cut of [Formula: see text] is defined as a set [Formula: see text] of edges (vertices) of [Formula: see text] whose removal disconnects [Formula: see text] leaving behind components of minimum degree at least [Formula: see text] The characterization of a minimum [Formula: see text]-vertex cut of the [Formula: see text]-dimensional hypercube [Formula: see text] is known. In this paper, we characterize a minimum [Formula: see text]-edge cut of [Formula: see text] Also, we obtain a sharp lower bound on the number of vertices of an [Formula: see text]-edge cut of [Formula: see text] and obtain some consequences.
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