科研之友
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摘要
Jaeger et al.[J. Combin. Theory, Ser B, 56 (1992) 165-182] conjectured that ever 3-edge-connected graph is Z(5)-connected. Let G be a 3-edge-connected simple graph on n vertices and A an abelian group with vertical bar A vertical bar >= 3. If a graph G* is obtained by repeatedly contracting nontrivial A-connected subgraphs of G until no such a subgraph left, we say G can be A-reduced to C*. It is proved in this paper that G is A-connected with vertical bar A vertical bar >= 5 if one of the following holds: (i) n <= 15; (ii) n = 16 and Delta >= 4; or (iii) n = 17 and Delta >= 5. As applications, we also show the following results. (1) For vertical bar A vertical bar >= 5 and n >= 17, if vertical bar E(G)vertical bar >= ((n-15)(2)) + 31, then G is A-connected. (2) For For vertical bar A vertical bar >= 4 and n >= 13, if vertical bar E(G)vertical bar >= ((n-11)(2)) + 23, then either G is A-connected or G can be A-reduced to the Petersen graph.
