共查询到20条相似文献,搜索用时 62 毫秒
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Liming Xiong 《Discrete Mathematics》2008,308(23):5891-5894
An even factor of a graph is a spanning subgraph of G in which all degrees are even, positive integers. In this paper, we characterize the claw-free graphs having even factors and then prove that the n-iterated line graph Ln(G) of G has an even factor if and only if every end branch of G has length at most n and every odd branch-bond of G has a branch of length at most n+1. 相似文献
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Thomas Böhme 《Discrete Mathematics》2006,306(7):666-669
We prove that for every graph H with the minimum degree δ?5, the third iterated line graph L3(H) of H contains as a minor. Using this fact we prove that if G is a connected graph distinct from a path, then there is a number kG such that for every i?kG the i-iterated line graph of G is -linked. Since the degree of Li(G) is even, the result is best possible. 相似文献
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Connectivity of iterated line graphs 总被引:1,自引:0,他引:1
Yehong Shao 《Discrete Applied Mathematics》2010,158(18):2081-2087
Let k≥0 be an integer and Lk(G) be the kth iterated line graph of a graph G. Niepel and Knor proved that if G is a 4-connected graph, then κ(L2(G))≥4δ(G)−6. We show that the connectivity of G can be relaxed. In fact, we prove in this note that if G is an essentially 4-edge-connected and 3-connected graph, then κ(L2(G))≥4δ(G)−6. Similar bounds are obtained for essentially 4-edge-connected and 2-connected (1-connected) graphs. 相似文献
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Mohammad Ghebleh 《Discrete Mathematics》2008,308(1):144-147
The line index of a graph G is the smallest k such that the kth iterated line graph of G is nonplanar. We show that the line index of a graph is either infinite or it is at most 4. Moreover, we give a full characterization of all graphs with respect to their line index. 相似文献
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Zhao Guangfu 《Journal of Graph Theory》1988,12(2):285-294
The main results of this paper are two characterizations of second iterated line graphs, i.e., two sets of necessary and sufficient conditions for the existence of solutions to the graph equation H = L2(G). A method to get a root of the graph equation is also given if one exists. 相似文献
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Xuding Zhu 《Discrete Mathematics》2009,309(18):5562-5568
Given a graph G and a positive integer p, χp(G) is the minimum number of colours needed to colour the vertices of G so that for any i≤p, any subgraph H of G of tree-depth i gets at least i colours. This paper proves an upper bound for χp(G) in terms of the k-colouring number of G for k=2p−2. Conversely, for each integer k, we also prove an upper bound for in terms of χk+2(G). As a consequence, for a class K of graphs, the following two statements are equivalent:
- (a)
- For every positive integer p, χp(G) is bounded by a constant for all G∈K.
- (b)
- For every positive integer k, is bounded by a constant for all G∈K.
- (c)
- For every positive integer q, ∇q(G) (the greatest reduced average density of G with rank q) is bounded by a constant for all G∈K.
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Akira Saito 《Discrete Mathematics》2009,309(16):5000-1723
We consider 2-factors with a bounded number of components in the n-times iterated line graph Ln(G). We first give a characterization of graph G such that Ln(G) has a 2-factor containing at most k components, based on the existence of a certain type of subgraph in G. This generalizes the main result of [L. Xiong, Z. Liu, Hamiltonian iterated line graphs, Discrete Math. 256 (2002) 407-422]. We use this result to show that the minimum number of components of 2-factors in the iterated line graphs Ln(G) is stable under the closure operation on a claw-free graph G. This extends results in [Z. Ryjá?ek, On a closure concept in claw-free graphs, J. Combin. Theory Ser. B 70 (1997) 217-224; Z. Ryjá?ek, A. Saito, R.H. Schelp, Closure, 2-factors and cycle coverings in claw-free graphs, J. Graph Theory 32 (1999) 109-117; L. Xiong, Z. Ryjá?ek, H.J. Broersma, On stability of the hamiltonian index under contractions and closures, J. Graph Theory 49 (2005) 104-115]. 相似文献
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B. Devadas Acharya 《Discrete Mathematics》1982,41(2):115-122
In this paper, the problem of determining graphs which are switching equivalent to at least one of their iterated line graphs is considered, and such connected graphs are characterized. 相似文献
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Toru Hasunuma 《Discrete Applied Mathematics》2007,155(9):1141-1154
In this paper, we study queue layouts of iterated line directed graphs. A k-queue layout of a directed graph consists of a linear ordering of the vertices and an assignment of each arc to exactly one of the k queues so that any two arcs assigned to the same queue do not nest. The queuenumber of a directed graph is the minimum number of queues required for a queue layout of the directed graph.We present upper and lower bounds on the queuenumber of an iterated line directed graph Lk(G) of a directed graph G. Our upper bound depends only on G and is independent of the number of iterations k. Queue layouts can be applied to three-dimensional drawings. From the results on the queuenumber of Lk(G), it is shown that for any fixed directed graph G, Lk(G) has a three-dimensional drawing with O(n) volume, where n is the number of vertices in Lk(G). These results are also applied to specific families of iterated line directed graphs such as de Bruijn, Kautz, butterfly, and wrapped butterfly directed graphs. In particular, the queuenumber of k-ary butterfly directed graphs is determined if k is odd. 相似文献
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The total interval number of an n-vertex graph with maximum degree Δ is at most (Δ + 1/Δ)n/2, with equality if and only if every component of the graph is KΔ,Δ. If the graph is also required to be connected, then the maximum is Δn/2 + 1 when Δ is even, but when Δ is odd it exceeds [Δ + 1/(2.5Δ + 7.7)]n/2 for infinitely many n. © 1997 John Wiley & Sons, Inc. J Graph Theory 25: 79–84, 1997 相似文献
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Ralph Faudree Ronald J. Gould Linda Lesniak Terri Lindquester 《Journal of Graph Theory》1995,19(3):397-409
We consider a generalized degree condition based on the cardinality of the neighborhood union of arbitrary sets of r vertices. We show that a Dirac-type bound on this degree in conjunction with a bound on the independence number of a graph is sufficient to imply certain hamiltonian properties in graphs. For K1,m-free grphs we obtain generalizations of known results. In particular we show: Theorem. Let r ≥ 1 and m ≥ 3 be integers. Then for each nonnegative function f(r, m) there exists a constant C = C(r, m, f(r, m)) such that if G is a graph of order n (n ≥ r, n > m) with δr(G) ≥ (n/3) + C and β (G) ≥ f(r, m), then (a) G is traceable if δ(G) ≥ r and G is connected; (b) G is hamiltonian if δ(G) ≥ r + 1 and G is 2-connected; (c) G is hamiltonian-connected if δ(G) ≥ r + 2 and G is 3-connected. © 1995 John Wiley & Sons, Inc. 相似文献
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Let be a graph. Denote by its -iterated line graph and denote by its Wiener index. Dobrynin and Melnikov conjectured that there exists no nontrivial tree and , such that . We prove this conjecture for trees which are not homeomorphic to the claw and , where is a tree consisting of 6 vertices, 2 of which have degree 3. 相似文献
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Paul A. Catlin Iqbalunnisa T. N. Janakiraman N. Srinivasan 《Journal of Graph Theory》1990,14(3):347-364
Let G be an undirected connected graph that is not a path. We define h(G) (respectively, s(G)) to be the least integer m such that the iterated line graph Lm(G) has a Hamiltonian cycle (respectively, a spanning closed trail). To obtain upper bounds on h(G) and s(G), we characterize the least integer m such that Lm(G) has a connected subgraph H, in which each edge of H is in a 3-cycle and V(H) contains all vertices of degree not 2 in Lm(G). We characterize the graphs G such that h(G) — 1 (respectively, s(G)) is greater than the radius of G. 相似文献
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By Petersen's theorem, a bridgeless cubic multigraph has a 2-factor. Fleischner generalised this result to bridgeless multigraphs of minimum degree at least three by showing that every such multigraph has a spanning even subgraph. Our main result is that every bridgeless simple graph with minimum degree at least three has a spanning even subgraph in which every component has at least four vertices. We deduce that if G is a simple bridgeless graph with n vertices and minimum degree at least three, then its line graph has a 2-factor with at most max{1,(3n-4)/10} components. This upper bound is best possible. 相似文献