Graphs and Combinatorics (GRAPH COMBINATOR )

Publisher: Springer Verlag


Graphs and Combinatorics is an international journal devoted to research concerning all aspects of combinatorial mathematics. In addition to original research papers and survey articles the journal also features short communications research problems and announcements. As particular attention is being paid to rapid publication researchers not only rely on Graphs and Combinatorics to keep them informed of current developments but also use it as a forum for publicizing their own work. Graphs and Combinatorics covers: graph theory combinatorics

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    Graphs and Combinatorics website
  • Other titles
    Graphs and combinatorics (Online), Graphs & combinatorics
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    Document, Periodical, Internet resource
  • Document type
    Internet Resource, Computer File, Journal / Magazine / Newspaper

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Springer Verlag

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    • Articles in some journals can be made Open Access on payment of additional charge
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Publications in this journal

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    ABSTRACT: Let B(n) be the subset lattice of {1, 2, . . . , n}. Sperner's theorem states that the width of B(n) is equal to the size of its biggest level. There have been several elegant proofs of this result, including an approach that shows that B(n) has a symmetric chain partition. An-other famous result concerning B(n) is that its cover graph is hamilton-ian. Motivated by these ideas and by the Middle Two Levels conjecture, we consider posets that have the Hamiltonian Cycle–Symmetric Chain Partition (HC-SCP) property. A poset of width w has this property if its cover graph has a hamiltonian cycle which parses into w symmetric chains. We show that the subset lattices have the HC-SCP property, and we obtain this result as a special case of a more general treatment.
    Graphs and Combinatorics 11/2014; 30(6).
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    ABSTRACT: Let G = (V, E) be a connected graph. The hamiltonian index h(G) (Hamilton-connected index hc(G)) of G is the least nonnegative integer k for which the iterated line graph L k (G) is hamiltonian (Hamilton-connected). In this paper we show the following. (a) If |V(G)| ≥ k + 1 ≥ 4, then in G k , for any pair of distinct vertices {u, v}, there exists k internally disjoint (u, v)-paths that contains all vertices of G; (b) for a tree T, h(T) ≤ hc(T) ≤ h(T) + 1, and for a unicyclic graph G, h(G) ≤ hc(G) ≤ max{h(G) + 1, k′ + 1}, where k′ is the length of a longest path with all vertices on the cycle such that the two ends of it are of degree at least 3 and all internal vertices are of degree 2; (c) we also characterize the trees and unicyclic graphs G for which hc(G) = h(G) + 1.
    Graphs and Combinatorics 11/2014; 30(6).
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    ABSTRACT: In 2001, Kawarabayashi proved that for any odd integer k ≥ 3, if a k-connected graph G is \({K^{-}_{4}}\) -free, then G has a k-contractible edge. He pointed out, by a counterexample, that this result does not hold when k is even. In this paper, we have proved the following two results on the subject: (1) For any even integer k ≥ 4, if a k-connected graph G is \({K_{4}^{-}}\) -free and d G (x) + d G (y) ≥ 2k + 1 hold for every two adjacent vertices x and y of V(G), then G has a k-contractible edge. (2) Let t ≥ 3, k ≥ 2t – 1 be integers. If a k-connected graph G is \({(K_{1}+(K_{2} \cup K_{1, t}))}\) -free and d G (x) + d G (y) ≥ 2k + 1 hold for every two adjacent vertices x and y of V(G), then G has a k-contractible edge.
    Graphs and Combinatorics 11/2014; 30(6).
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    ABSTRACT: A simple matrix is a (0,1)-matrix with no repeated columns. For a (0,1)-matrix F, we say that a (0,1)-matrix A has F as a configuration if there is a submatrix of A which is a row and column permutation of F (trace is the set system version of a configuration). Let \({\|A\|}\) denote the number of columns of A. We define \({{\rm forb}(m, F) = {\rm max}\{\|A\| \,:\, A}\) is m-rowed simple matrix and has no configuration F. We extend this to a family \({\mathcal{F} = \{F_1, F_2, \ldots , F_t\}}\) and define \({{\rm forb}(m, \mathcal{F}) = {\rm max}\{\|A\| \,:\, A}\) is m-rowed simple matrix and has no configuration \({F \in \mathcal{F}\}}\) . We consider products of matrices. Given an m 1 × n 1 matrix A and an m 2 × n 2 matrix B, we define the product A × B as the (m 1 + m 2) × n 1n 2 matrix whose columns consist of all possible combinations obtained from placing a column of A on top of a column of B. Let I k denote the k × k identity matrix, let \({I_k^{c}}\) denote the (0,1)-complement of I k and let T k denote the k × k upper triangular (0,1)-matrix with a 1 in position i, j if and only if i ≤ j. We show forb(m, {I 2 × I 2, T 2 × T 2}) is \({\Theta(m^{3/2})}\) while obtaining a linear bound when forbidding all 2-fold products of all 2 × 2 (0,1)-simple matrices. For two matrices F, P, where P is m-rowed, let \({f(F, P) = {\rm max}_{A} \{\|A\| \,:\,A}\) is m-rowed submatrix of P with no configuration F}. We establish f(I 2 × I 2, I m/2 × I m/2) is \({\Theta(m^{3/2})}\) whereas f(I 2 × T 2, I m/2 × T m/2) and f(T 2 × T 2, T m/2 × T m/2) are both \({\Theta(m)}\) . Additional results are obtained. One of the results requires extensive use of a computer program. We use the results on patterns due to Marcus and Tardos and generalizations due to Klazar and Marcus, Balogh, Bollobás and Morris.
    Graphs and Combinatorics 11/2014; 30(6).
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    ABSTRACT: For bipartite graphs G 1 ,G 2 ,⋯,G k , the bipartite Ramsey number b(G 1 ,G 2 ,⋯,G k ) is the least positive integer b so that any colouring of the edges of K b ,b with k colours will result in a copy of G i in the ith colour for some i. A tree of diameter three is called a bistar, and will be denoted by B(s,t), where s≥2 and t≥2 are the degrees of the two support vertices. In this paper we will obtain some exact values for b(B(s,t),B(s,t)) and b(B(s,s),B(s,s)). Furtermore, we will show that if k colours are used, with k≥2 and s≥2, then b k (B(s,s))≤⌈k(s-1)+(s-1) 2 (k 2 -k)-k(2s-4)⌉. Finally, we show that for s≥3 and k≥2, the Ramsey number r k (B(s,s))≤⌈2k(s-1)+1 2+1 2(4k(s-1)+1) 2 -8k(2s 2 -s-2)⌉·
    Graphs and Combinatorics 09/2014;
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    ABSTRACT: In this paper, we present a sharp upper bound for the least signless Laplacian eigenvalue of a graph involving its domination number. Moreover, we determine some extremal graphs which attain the sharp bound.
    Graphs and Combinatorics 09/2014;
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    ABSTRACT: Lai, Shao and Zhan (J Graph Theory 48:142–146, 2005) showed that every 3-connected N 2-locally connected claw-free graph is Hamiltonian. In this paper, we generalize this result and show that every 3-connected claw-free graph G such that every locally disconnected vertex lies on some induced cycle of length at least 4 with at most 4 edges contained in some triangle of G is Hamiltonian. It is best possible in some sense.
    Graphs and Combinatorics 09/2014; 30(5).
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    ABSTRACT: In this paper we obtain criteria which allow us to decide, for a large class of graphs, whether they are hyperbolic or not. We are especially interested in the planar graphs which are the ``boundary" (the $1$-skeleton) of a tessellation of the Euclidean plane. Furthermore, we prove that a graph obtained as the $1$-skeleton of a general CW $2$-complex is hyperbolic if and only if its dual graph is hyperbolic.
    Graphs and Combinatorics 08/2014;
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    ABSTRACT: The automorphic H-chromatic index of a graph Γ is the minimum integer m for which Γ has a proper edge-coloring with m colors preserved by a given subgroup H of the full automorphism group of Γ. We determine upper bounds for this index in terms of the chromatic index of Γ for some abelian 2-groups H.
    Graphs and Combinatorics 07/2014;
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    ABSTRACT: We consider only digraphs that are oriented graphs, meaning orientations of simple finite graphs. An oriented graph $D=(V,A)$ with minimum degree $d$ is called $d$-arc-dominated if for every arc $(x,y) \in A$ there is a vertex $u \in V$ with outdegree $d$ such that both $(u,x) \in A$ and $(u,y) \in A$ hold. In this paper, we show that for any integer $d \ge 3$ the girth of a $d$-arc-dominated oriented graph is less than or equal to $d$. Moreover, for every integer $t$ with $3 \le t \le d$ there is a $d$-arc-dominated oriented graph with girth $t$. We also give a characterization for oriented graphs with both minimum outdegree and girth $d$ to be $d$-arc-dominated and classify all $d$-arc-dominated $d$-circular oriented graphs with girth $d$.
    Graphs and Combinatorics 07/2014; 30(4):1045 - 1054.
  • Graphs and Combinatorics 06/2014;
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    ABSTRACT: The square $G^2$ of a graph $G$ is defined on the vertex set $V(G)$ of $G$ such that any two vertices with distance at most two in $G$ are linked by an edge. In this paper, the chromatic number and equitable chromatic number of the square $S^2(n,k)$ of Sierpiński graph $S(n,k)$ are studied. It is obtained that $\chi (S^2(n,k))=\chi _{=}(S^2(n,k))=k+1$ for $n\ge 2$ and $k\ge 2$ .
    Graphs and Combinatorics 06/2014;
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    ABSTRACT: Let k be a non-negative integer. A branch vertex of a tree is a vertex of degree at least three. We show two sufficient conditions for a connected claw-free graph to have a spanning tree with a bounded number of branch vertices: (i) A connected claw-free graph has a spanning tree with at most k branch vertices if its independence number is at most 2k + 2. (ii) A connected claw-free graph of order n has a spanning tree with at most one branch vertex if the degree sum of any five independent vertices is at least n − 2. These conditions are best possible. A related conjecture also is proposed.
    Graphs and Combinatorics 03/2014;
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    ABSTRACT: Three intersection theorems are proved. First, we determine the size of the largest set system, where the system of the pairwise unions is l-intersecting. Then we investigate set systems where the union of any s sets intersect the union of any t sets. The maximal size of such a set system is determined exactly if s+t<5, and asymptotically if s+t>4. Finally, we exactly determine the maximal size of a k-uniform set system that has the above described (s,t)-union-intersecting property, for large enough n.
    Graphs and Combinatorics 03/2014;
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    ABSTRACT: The Cycle Double Cover Conjecture claims that every bridgeless graph has a cycle double cover and the Strong Cycle Double Conjecture states that every such graph has a cycle double cover containing any specified circuit. In this paper, we get a necessary and sufficient condition for bridgeless graphs to have a strong 5-cycle double cover. Similar condition for the existence of 5-cycle double covers is also obtained. These conditions strengthen/improve some known results.
    Graphs and Combinatorics 01/2014;
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    ABSTRACT: A plane graph G is edge-face k-colorable if the elements of ${E(G) \cup F(G)}$ can be colored with k colors so that any two adjacent or incident elements receive different colors. Sanders and Zhao conjectured that every plane graph with maximum degree Δ is edge-face (Δ + 2)-colorable and left the cases ${\Delta \in \{4, 5, 6\}}$ unsolved. In this paper, we settle the case Δ = 6. More precisely, we prove that every plane graph with maximum degree 6 is edge-face 8-colorable.
    Graphs and Combinatorics 01/2014;