Packing spanning trees in highly essentially connected graphs.
Discrete Mathematics(2019)
摘要
Let τ(G) be the maximum number of mutually edge-disjoint spanning trees contained in a graph G and let κ′(G) denote the edge-connectivity of G. As a corollary of the spanning trees packing theorem by Nash-Williams and Tutte, it is known that if κ′(G)≥2k, then τ(G)≥k. An edge-cut X of G is an essential edge-cut if G−X contains at least two nontrivial components; and G is essentially k-edge-connected if G does not have an essential edge-cut of size less than k. In this paper, we prove that every g-edge-connected, essentially h-edge-connected graph G with g≥k+1 and h≥g2g−k−2 satisfies τ(G)≥k. This result is sharp in the sense that there exist infinitely many graphs showing that neither inequality in the hypothesis can be relaxed. Applications to circular flows of graphs, spanning connectivity of line graphs and supereulerian width of graphs are discussed. In particular, we obtained the following, for given integers g and k with k>1 and 2k−1≥g≥k+1.
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关键词
Edge-disjoint spanning trees,Essential edge connectivity,Nowhere-zero flows,Circular flow number,Spanning connectivity,Supereulerian width
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