A graph G is said to be super-connected or simply super-k,if each minimum vertex cut of G isolates a vertex.A graph G is said to be a k-vertex-orbit graph if there are k vertex orbits when Aut(G)acts on V(G).A graph G...A graph G is said to be super-connected or simply super-k,if each minimum vertex cut of G isolates a vertex.A graph G is said to be a k-vertex-orbit graph if there are k vertex orbits when Aut(G)acts on V(G).A graph G is said to be a k-edge-orbit graph if there are k edge orbits when Aut(G)acts on edge set E(G).In this paper,we give a necessary and sufficient condition for connected bipartite 2-vertex-orbit graphs to be super-k.For 2-edge-orbit graphs,we give a sufficient condition for connected 2-edge-orbit graphs to be super-k.In addition,we show that if G is a k-regular connected irreducible II-kind 2-edge-orbit graph with k≤6 and girth g(G)≥6,or G is a k-regular connected irreducible III-kind 2-edge-orbit graph with k≤6 and girth g(G)≥8,then G is super-connected.展开更多
基金Supported by the National Natural Science Foundation of Xinjiang(2020D04046)the National Natural Science Foundation of Shanxi(20210302123097)the National Natural Science Foundation of China(12371356,11961067).
文摘A graph G is said to be super-connected or simply super-k,if each minimum vertex cut of G isolates a vertex.A graph G is said to be a k-vertex-orbit graph if there are k vertex orbits when Aut(G)acts on V(G).A graph G is said to be a k-edge-orbit graph if there are k edge orbits when Aut(G)acts on edge set E(G).In this paper,we give a necessary and sufficient condition for connected bipartite 2-vertex-orbit graphs to be super-k.For 2-edge-orbit graphs,we give a sufficient condition for connected 2-edge-orbit graphs to be super-k.In addition,we show that if G is a k-regular connected irreducible II-kind 2-edge-orbit graph with k≤6 and girth g(G)≥6,or G is a k-regular connected irreducible III-kind 2-edge-orbit graph with k≤6 and girth g(G)≥8,then G is super-connected.
