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The strong matching preclusion number of a graph is the minimum number of vertices and edges whose deletion results in a graph that has neither perfect matchings nor almost-perfect matchings. Park and Ihm introduced the problem of strong matching preclusion under the condition that no isolated vertex is created as a result of faults. In this paper, we find the conditional strong matching preclusion number for the n-dimensional alternating group graph AG n .

Pancylicity was introduced by Bondy in 1971. A graph G with vertex set V(G) and edge set E(G) is pancyclic if it contains cycles of lengths l, for 3 ≤ l ≤ |V(G)|. This concept has been extended to edge-pancyclicity. If every edge of G is in a cycle of every length, G is edge-pancyclic. If every edge lies on cycles of all lengths ranging from k to |V(G)|, G is k-edge-pancyclic. In this paper, we prove that the n-dimensional pancake graph is 7-edge-pancyclic.

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