منابع مشابه
Spanning Trees in Regular Graphs
Let X be a regular graph with degree k 2 3 and order n. Then the number of spanning trees of X is where yk, ck and (3,,k(l/k) are positive constants, and p, is the number of equivalence classes of certain closed walks of length i in X. The value (k-l)k-l Ck = (k2_2k)"i/2)-l is shown to be the best possible in the sense that K(x~)"" +c^ for some increasing sequence X,, X2,.. . of regular graphs ...
متن کاملSpanning Star Trees in Regular Graphs
For a subset W of vertices of an undirected graph G, let S(W ) be the subgraph consisting of W , all edges incident to at least one vertex in W , and all vertices adjacent to at least one vertex in W . If S(W ) is a tree containing all the vertices of G, then we call it a spanning star tree of G. In this case W forms a weakly connected but strongly acyclic dominating set for G. We prove that fo...
متن کاملEdge-disjoint spanning trees and eigenvalues of regular graphs
Partially answering a question of Paul Seymour, we obtain a sufficient eigenvalue condition for the existence of k edge-disjoint spanning trees in a regular graph, when k ∈ {2, 3}. More precisely, we show that if the second largest eigenvalue of a d-regular graph G is less than d − 2k−1 d+1 , then G contains at least k edge-disjoint spanning trees, when k ∈ {2, 3}. We construct examples of grap...
متن کاملCompletely Independent Spanning Trees in Some Regular Graphs
Let k ≥ 2 be an integer and T1, . . . , Tk be spanning trees of a graph G. If for any pair of vertices (u, v) of V (G), the paths from u to v in each Ti, 1 ≤ i ≤ k, do not contain common edges and common vertices, except the vertices u and v, then T1, . . . , Tk are completely independent spanning trees in G. For 2k-regular graphs which are 2k-connected, such as the Cartesian product of a compl...
متن کاملSpanning Trees with Many Leaves in Regular Bipartite Graphs
Given a d-regular bipartite graph Gd, whose nodes are divided in black nodes and white nodes according to the partition, we consider the problem of computing the spanning tree of Gd with the maximum number of black leaves. We prove that the problem is NP hard for any fixed d ≥ 4 and we present a simple greedy algorithm that gives a constant approximation ratio for the problem. More precisely ou...
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ژورنال
عنوان ژورنال: Časopis pro pěstování matematiky
سال: 1970
ISSN: 0528-2195
DOI: 10.21136/cpm.1970.108329