共查询到20条相似文献,搜索用时 0 毫秒
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A spanning tree of a properly edge-colored complete graph, , is rainbow provided that each of its edges receives a distinct color. In 1996, Brualdi and Hollingsworth conjectured that if is properly -edge-colored, then the edges of can be partitioned into rainbow spanning trees except when . By means of an explicit, constructive approach, in this paper we construct mutually edge-disjoint rainbow spanning trees for any positive value of . Not only are the rainbow trees produced, but also some structure of each rainbow spanning tree is determined in the process. This improves upon best constructive result to date in the literature which produces exactly three rainbow trees. 相似文献
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Suh-Ryung Kim 《Discrete Applied Mathematics》2008,156(18):3522-3524
For a graph G, it is known to be a hard problem to compute the competition number k(G) of the graph G in general. In this paper, we give an explicit formula for the competition numbers of complete tripartite graphs. 相似文献
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Maria Axenovich Heiko Harborth Arnfried Kemnitz Meinhard Möller Ingo Schiermeyer 《Graphs and Combinatorics》2007,23(2):123-133
Let Qn be a hypercube of dimension n, that is, a graph whose vertices are binary n-tuples and two vertices are adjacent iff the corresponding n-tuples differ in exactly one position. An edge coloring of a graph H is called rainbow if no two edges of H have the same color. Let f(G,H) be the largest number of colors such that there exists an edge coloring of G with f(G,H) colors such that no subgraph isomorphic to H is rainbow. In this paper we start the investigation of this anti-Ramsey problem by providing bounds on f(Qn,Qk) which are asymptotically tight for k = 2 and by giving some exact results. 相似文献
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It is proved here that any edge-coloring critical graph of order n and maximum degree Δ?8 has the size at least 3(n+Δ−8). It generalizes a result of Hugh Hind and Yue Zhao. 相似文献
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《Discrete Mathematics》2020,343(1):111629
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Michael J. Fisher 《Discrete Mathematics》2008,308(11):2240-2246
We determine the values of s and t for which there is a coloring of the edges of the complete bipartite graph Ks,t which admits only the identity automorphism. In particular, this allows us to determine the distinguishing number of the Cartesian product of complete graphs. 相似文献
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For a finite simple edge-colored connected graph G (the coloring may not be proper), a rainbow path in G is a path without two edges colored the same; G is rainbow connected if for any two vertices of G, there is a rainbow path connecting them. Rainbow connection number, rc(G), of G is the minimum number of colors needed to color its edges such that G is rainbow connected. Chakraborty et al. (2011) [5] proved that computing rc(G) is NP-hard and deciding if rc(G)=2 is NP-complete. When edges of G are colored with fixed number k of colors, Kratochvil [6] proposed a question: what is the complexity of deciding whether G is rainbow connected? is this an FPT problem? In this paper, we prove that any maximal outerplanar graph is k rainbow connected for suitably large k and can be given a rainbow coloring in polynomial time. 相似文献
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On the use of graphs in discrete tomography 总被引:2,自引:2,他引:0
Dominique de Werra Marie-Christine Costa Christophe Picouleau Bernard Ries 《4OR: A Quarterly Journal of Operations Research》2008,6(2):101-123
In this tutorial paper, we consider the basic image reconstruction problem which stems from discrete tomography. We derive
a graph theoretical model and we explore some variations and extensions of this model. This allows us to establish connections
with scheduling and timetabling applications. The complexity status of these problems is studied and we exhibit some polynomially
solvable cases. We show how various classical techniques of operations research like matching, 2-SAT, network flows are applied
to derive some of these results.
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Carl Johan Casselgren 《Discrete Mathematics》2011,(13):168
Let Ks×m be the complete multipartite graph with s parts and m vertices in each part. Assign to each vertex v of Ks×m a list L(v) of colors, by choosing each list uniformly at random from all 2-subsets of a color set C of size σ(m). In this paper we determine, for all fixed s and growing m, the asymptotic probability of the existence of a proper coloring φ, such that φ(v)∈L(v) for all v∈V(Ks×m). We show that this property exhibits a sharp threshold at σ(m)=2(s−1)m. 相似文献
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Ohba has conjectured that if G is a k-chromatic graph with at most 2k+1 vertices, then the list chromatic number or choosability of G is equal to its chromatic number χ(G), which is k. It is known that this holds if G has independence number at most three. It is proved here that it holds if G has independence number at most five. In particular, and equivalently, it holds if G is a complete k-partite graph and each part has at most five vertices. 相似文献
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The tree partition number of an r‐edge‐colored graph G, denoted by tr(G), is the minimum number k such that whenever the edges of G are colored with r colors, the vertices of G can be covered by at most k vertex‐disjoint monochromatic trees. We determine t2(K(n1, n2,…, nk)) of the complete k‐partite graph K(n1, n2,…, nk). In particular, we prove that t2(K(n, m)) = ? (m‐2)/2n? + 2, where 1 ≤ n ≤ m. © 2004 Wiley Periodicals, Inc. J Graph Theory 48: 133–141, 2005 相似文献
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Brualdi and Hollingsworth conjectured in Brualdi and Hollingsworth (1996) that in any complete graph , , which is properly colored with colors, the edge set can be partitioned into edge disjoint rainbow spanning trees (where a graph is said to be rainbow if its edges have distinct colors). Constantine (2002) strengthened this conjecture asking the rainbow spanning trees to be pairwise isomorphic. He also showed an example satisfying his conjecture for every . Caughmann, Krussel and Mahoney (2017) recently showed a first infinite family of edge colorings for which the conjecture of Brualdi and Hollingsworth can be verified. In the present paper, we extend this result to all edge-colorings arising from cyclic 1-factorizations of constructed by Hartman and Rosa (1985). Finally, we remark that our constructions permit to extend Constatine’s result also to all . 相似文献
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For a given graph H and a positive n, the rainbow number ofH, denoted by rb(n,H), is the minimum integer k so that in any edge-coloring of Kn with k colors there is a copy of H whose edges have distinct colors. In 2004, Schiermeyer determined rb(n,kK2) for all n≥3k+3. The case for smaller values of n (namely, ) remained generally open. In this paper we extend Schiermeyer’s result to all plausible n and hence determine the rainbow number of matchings. 相似文献
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