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A sequence d=(d1,d2,…,dn) is graphic if there is a simple graph G with degree sequence d, and such a graph G is called a realization of d. A graphic sequence d is line-hamiltonian if d has a realization G such that L(G) is hamiltonian, and is supereulerian if d has a realization G with a spanning eulerian subgraph. In this paper, it is proved that a nonincreasing graphic sequence d=(d1,d2,…,dn) has a supereulerian realization if and only if dn≥2 and that d is line-hamiltonian if and only if either d1=n−1, or ∑di=1di≤∑dj≥2(dj−2). 相似文献
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Alexandru Ioan Tomescu 《Discrete Applied Mathematics》2008,156(1):125-130
In this paper characterizations of connected unicyclic and bicyclic graphs in terms of the degree sequence, as well as the graphs in these classes minimal with respect to the degree distance are given. 相似文献
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A note on degree sum conditions for 2-factors with a prescribed number of cycles in bipartite graphs
Let be a balanced bipartite graph of order , and let be the minimum degree sum of two non-adjacent vertices in different partite sets of . In 1963, Moon and Moser proved that if , then is hamiltonian. In this note, we show that if is a positive integer, then the Moon–Moser condition also implies the existence of a 2-factor with exactly cycles for sufficiently large graphs. In order to prove this, we also give a condition for the existence of vertex-disjoint alternating cycles with respect to a chosen perfect matching in . 相似文献
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Let G be the circuit graph of any connected matroid M with minimum degree 5(G). It is proved that its connectivity κ(G) ≥2|E(M) - B(M)| - 2. Therefore 5(G) ≥ 2|E(M) - B(M)| - 2 and this bound is the best possible in some sense. 相似文献
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设G是无向无环的有限图 ,若G有一个生成子图是欧拉图 (Euler) ,则称G是超欧拉图 (Supereulerian) .本文不利用收缩方法 ,直接证明了 :当图G至多差一边有两棵边不相交的生成树时 ,G是超欧拉图或者G有割边 . 相似文献
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This note proves a conjecture of Merris that the minimal value of entries of the doubly stochastic matrix of the degree antiregular graph En of order n ≥ 3 is equal to (l/2(n + l)). 相似文献
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This note proves a conjecture of Merris that the minimal value of entries of the doubly stochastic matrix of the degree antiregular graph En of order n ≥ 3 is equal to (l/2(n + l)). 相似文献
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Haruhide Matsuda 《Discrete Mathematics》2009,309(11):3653-3658
We give sufficient conditions for a graph to have degree bounded trees. Let G be a connected graph and A a vertex subset of G. We denote by σk(A) the minimum value of the degree sum in G of any k independent vertices in A and by w(G−A) the number of components in the induced subgraph G−A. Our main results are the following: (i) If σk(A)≥|V(G)|−1, then G contains a tree T with maximum degree at most k and A⊆V(T). (ii) If σk−w(G−A)(A)≥|A|−1, then G contains a spanning tree T such that dT(x)≤k for every x∈A. These are generalizations of the result by Win [S. Win, Existenz von Gerüsten mit Vorgeschriebenem Maximalgrad in Graphen, Abh. Math. Sem. Univ. Hamburg 43 (1975) 263-267] and the degree conditions are sharp. 相似文献
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Florian Pfender 《Discrete Mathematics》2009,309(9):2922-2924
We show that line graphs G=L(H) with σ2(G)≥7 contain cycles of all lengths k, 2rad(H)+1≤k≤c(G). This implies that every line graph of such a graph with 2rad(H)≥Δ(H) is subpancyclic, improving a recent result of Xiong and Li. The bound on σ2(G) is best possible. 相似文献
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In this paper, we focus on the directed minimum degree spanning tree problem and the minimum time broadcast problem. Firstly, we propose a polynomial time algorithm for the minimum degree spanning tree problem in directed acyclic graphs. The algorithm starts with an arbitrary spanning tree, and iteratively reduces the number of vertices of maximum degree. We can prove that the algorithm must reduce a vertex of the maximum degree for each phase, and finally result in an optimal tree. The algorithm terminates in O(mnlogn) time, where m and n are the numbers of edges and vertices of the graph, respectively. Moreover, we apply the new algorithm to the minimum time broadcast problem. Two consequences for directed acyclic graphs are: (1) the problem under the vertex-disjoint paths mode can be approximated within a factor of of the optimum in O(mnlogn)-time; (2) the problem under the edge-disjoint paths mode can be approximated within a factor of O(Δ*/logΔ*) of the optimum in O(mnlogn)-time, where Δ* is the minimum k such that there is a spanning tree of the graph with maximum degree k. 相似文献
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The neighborhood degree list (NDL) is a graph invariant that refines information given by the degree sequence and joint degree matrix of a graph and is useful in distinguishing graphs having the same degree sequence. We show that the space of realizations of an NDL is connected via a switching operation. We then determine the NDLs that have a unique realization by a labeled graph; the characterization ties these NDLs and their realizations to the threshold graphs and difference graphs. 相似文献
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Zbigniew R. Bogdanowicz 《Discrete Mathematics》2009,309(10):3074-3082
We show that for positive integers n, m with n(n−1)/2≥m≥n−1, the graph Ln,m having n vertices and m edges that consists of an (n−k)-clique and k−1 vertices of degree 1 has the fewest spanning trees among all connected graphs on n vertices and m edges. This proves Boesch’s conjecture [F.T. Boesch, A. Satyanarayana, C.L. Suffel, Least reliable networks and reliability domination, IEEE Trans. Commun. 38 (1990) 2004-2009]. 相似文献
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Graph coloring is an important tool in the study of optimization,computer science,network design,e.g.,file transferring in a computer network,pattern matching,computation of Hessians matrix and so on.In this paper,we consider one important coloring,vertex coloring of a total graph,which is also called total coloring.We consider a planar graph G with maximum degree Δ(G)≥8,and proved that if G contains no adjacent i,j-cycles with two chords for some i,j∈{5,6,7},then G is total-(Δ+1)-colorable. 相似文献
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Ioan Tomescu 《Discrete Mathematics》2009,309(9):2745-788
The degree distance of a connected graph, introduced by Dobrynin, Kochetova and Gutman, has been studied in mathematical chemistry. In this paper some properties of graphs having minimum degree distance in the class of connected graphs of order n and size m≥n−1 are deduced. It is shown that any such graph G has no induced subgraph isomorphic to P4, contains a vertex z of degree n−1 such that G−z has at most one connected component C such that |C|≥2 and C has properties similar to those of G.For any fixed k such that k=0,1 or k≥3, if m=n+k and n≥k+3 then the extremal graph is unique and it is isomorphic to K1+(K1,k+1∪(n−k−3)K1). 相似文献