# Graphs and Combinatorics (GRAPH COMBINATOR )

## Description

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

## Impact factor 0.33

- Hide impact factor historyImpact factorYear
- 5-year impact0.42
- Cited half-life9.70
- Immediacy index0.14
- Eigenfactor0.00
- Article influence0.48
- WebsiteGraphs and Combinatorics website
- Other titlesGraphs and combinatorics (Online), Graphs & combinatorics
- ISSN0911-0119
- OCLC39980543
- Material typeDocument, Periodical, Internet resource
- Document typeInternet Resource, Computer File, Journal / Magazine / Newspaper

## Publisher details

- Pre-print
- Author can archive a pre-print version

- Post-print
- Author can archive a post-print version

- Conditions
- Author's pre-print on pre-print servers such as arXiv.org
- Author's post-print on author's personal website immediately
- Author's post-print on any open access repository after 12 months after publication
- Publisher's version/PDF cannot be used
- Published source must be acknowledged
- Must link to publisher version
- Set phrase to accompany link to published version (see policy)
- Articles in some journals can be made Open Access on payment of additional charge

- Classification green

## Publications in this journal

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**ABSTRACT:**In this paper, we characterize connected \(\{K_{1,3},N(2,1,0)\}\) -free but not \(N(1,1,1)\) -free graphs. By combining our result and a theorem showed by Duffus et al. (every \(2\) -connected \(\{K_{1,3},N(1,1,1)\}\) -free graph is Hamiltonian), we give an alternative proof of Bedrossian’s theorem (every \(2\) -connected \(\{K_{1,3},N(2,1,0)\}\) -free graph is Hamiltonian).Graphs and Combinatorics 01/2015; - [Show abstract] [Hide abstract]

**ABSTRACT:**For two given graphs \(G_1\) and \(G_2\) , the planar Ramsey number \(PR(G_1,G_2)\) is the smallest integer \(N\) such that for any planar graph \(G\) of order \(N\) , either \(G\) contains \(G_1\) or the complement of \(G\) contains \(G_2\) . Let \(K_m\) denote a complete graph of order \(m\) and \(W_n\) a wheel of order \(n+1\) . In this paper, we determine all planar Ramsey numbers \(PR(K_m,W_n)\) .Graphs and Combinatorics 01/2015; - [Show abstract] [Hide abstract]

**ABSTRACT:**We classify all possible zero-divisor graphs of a particular family of quotients of \(\mathbf{Z}_4[x,y,w,z]\) . As the 90 quotients vary, we obtain a total of 7 graphs, corresponding to seven isomorphism classes, and one of these graphs provides a new example which contradicts Beck’s conjecture on the chromatic number of a zero-divisor graph. The algebraic analysis is strongly supported by the combinatorial setting, as already shown in a previous paper, where the graph-theoretical tools were presented and successfully applied to \(\mathbf{Z}_4[x,y,z]\) —therefore, the just smaller case—in order to get a deeper knowledge of the classical counterexample to Beck’s conjecture.Graphs and Combinatorics 01/2015; - [Show abstract] [Hide abstract]

**ABSTRACT:**For a graph G, we denote by δ(G) the minimum degree of G. A graph G is said to be claw-free if G has no induced subgraph isomorphic to K 1, 3. In this article, we prove that every claw-free graph G with minimum degree at least 4 has a 2-factor in which each cycle contains at least \({\big\lceil\frac{\delta(G) - 1}{2}\big\rceil}\) vertices and every 2-connected claw-free graph G with minimum degree at least 3 has a 2-factor in which each cycle contains at least δ(G) vertices. For the case where G is 2-connected, the lower bound on the length of a cycle is best possible.Graphs and Combinatorics 01/2015; 31(1). - [Show abstract] [Hide abstract]

**ABSTRACT:**Let \({P\subset\mathbb{R}^{2}}\) be a set of n points, of which k lie in the interior of the convex hull CH(P) of P. Let us call a triangulation T of P even (odd) if and only if all its vertices have even (odd) degree, and pseudo-even (pseudo-odd) if at least the k interior vertices have even (odd) degree. On the one hand, triangulations having all its interior vertices of even degree have one nice property; their vertices can be 3-colored, see (Heawood in Quart J Pure Math 29:270–285, 1898, Steinberg in A source book for challenges and directions, vol 55. Elsevier, Amsterdam, pp 211–248, 1993, Diks et al. in Lecture notes in computer science, vol 2573. Springer, Berlin, pp 138–149, 2002). On the other hand, odd triangulations have recently found an application in the colored version of the classic “Happy Ending Problem” of Erdős and Szekeres, see (Aichholzer et al. in SIAM J Discrete Math 23(4):2147–2155, 2010). It is easy to prove that there are sets of points that admit neither pseudo-even nor pseudo-odd triangulations. In this paper we show nonetheless how to construct a set of Steiner points S = S(P) of size at most \({\frac{k}{3} + c}\) , where c is a positive constant, such that a pseudo-even (pseudo-odd) triangulation can be constructed on \({P \cup S}\) . Moreover, we also show that even (odd) triangulations can always be constructed using at most \({\frac{n}{3} + c}\) Steiner points, where again c is a positive constant. Our constructions have the property that all but at most two Steiner points lie in the interior of CH(P).Graphs and Combinatorics 01/2015; 31(1). - [Show abstract] [Hide abstract]

**ABSTRACT:**In this note, we consider the trees (caterpillars) that minimize the number of subtrees among trees with a given degree sequence. This is a question naturally related to the extremal structures of some distance based graph invariants. We first confirm the expected fact that the number of subtrees is minimized by some caterpillar. As with other graph invariants, the specific optimal caterpillar is nearly impossible to characterize and depends on the degree sequence. We provide some simple properties of such caterpillars as well as observations that will help finding the optimal caterpillar.Graphs and Combinatorics 01/2015; 31(1). -
##### Article: On Traceable Line Graphs

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**ABSTRACT:**Let G be a simple graph of order n and D 1(G) be the set of vertices of degree 1 in G. In this paper, we prove that if G − D 1(G) is 2-edge-connected and if for every edge \({xy \in E(G)}\) , max{d(x), d(y)} ≥ n/6−1, then for n large, L(G) is traceable with the exception of a class of well characterized graphs. A similar result in (Lai, Discrete Math 178:93–107, 1998) states that if we replace 6 by 5 in the above degree condition, then for n large, L(G) is Hamiltonian with the exception of a class of well characterized graphs.Graphs and Combinatorics 01/2015; 31(1). - [Show abstract] [Hide abstract]

**ABSTRACT:**Let \({\varepsilon_{0}}\) , \({\varepsilon_{1}}\) be two linear homogenous equations, each with at least three variables and coefficients not all the same sign. Define the 2-color off-diagonal Rado number \({R_2(\varepsilon_{0}, \varepsilon_{1})}\) to be the smallest integer N such that for any 2-coloring of [1, N], it must admit a monochromatic solution to \({\varepsilon_{0}}\) of the first color or a monochromatic solution to \({\varepsilon_{1}}\) of the second color. In this paper, we establish two exact formulas of R 2(3x + 3y = z, 3x + 3qy = z) and R 2(2x + 3y = z, 2x + 2qy = z).Graphs and Combinatorics 01/2015; 31(1). - [Show abstract] [Hide abstract]

**ABSTRACT:**Based on Bagina’s Proposition, it has previously been demonstrated that there remain 34 cases where it is uncertain whether a convex pentagon can generate an edge-to-edge tiling. In this paper, these cases are further refined by imposing extra edge conditions. To investigate the resulting 42 cases, the properties of convex pentagonal tiles that can generate an edge-to-edge tiling are identified. These properties are the key to generating a perfect list of the types of convex pentagonal tiles that can generate an edge-to-edge tiling.Graphs and Combinatorics 01/2015; 31(1). - [Show abstract] [Hide abstract]

**ABSTRACT:**The bandwidth is an important invariant in both theoretic and applied fields. The extremal graph problem on bandwidth is to determine the minimum size of a graph G with order n and bandwidth B, denoted by m(n, B). The results of m(n, n − 1) and m(n, n − 2) have been known in the literature. This paper studies m(n, n − 3) as well as the extremal graphs. In particular, we concentrate on a relation between m(n, n − 3) and ex(n, C 4), i.e., the maximum size of a graph without 4-cycles. The latter is a well-known open problem proposed by P. Erdös more than 70 years ago.Graphs and Combinatorics 01/2015; 31(1). - [Show abstract] [Hide abstract]

**ABSTRACT:**The signed graph \({(G,\sigma)}\) is a graph G = (V, E) with a signature \({\sigma}\) from E to sign group {+, −}. A signed graph is orientation embedded in a surface when it is 2-cellular embedded so that the positive cycles are orientation preserving and the negative cycles are orientation reversing. The demigenus of a signed graph \({(G,\sigma)}\) is the minimal Euler genus over all surfaces S in which \({(G,\sigma)}\) can be orientation embedded. Finding the largest demigenus over all signatures on the complete bipartite graph K m,n is an open problem (Archdeacon, in Problems in topological graph theory, an ongoing online list of open problems, http://www.cems.uvm.edu/~darchdea/problems/signknm.htm). In this paper, by introducing diamond product for signed graphs, the largest demigenus over all signatures on the complete bipartite graph K 3,n is determined.Graphs and Combinatorics 01/2015; 31(1). - [Show abstract] [Hide abstract]

**ABSTRACT:**Given an undirected graph $G = (V,E)$ and two positive integers $k$ and $d$, we are interested in finding a set of edges (resp. non-edges) of size at most $k$ to delete (resp. to add) in such a way that the chromatic number (resp. stability number) in the resulting graph will decrease by at least $d$ compared to the original graph. We investigate these two problems in various classes of graphs (split graphs, threshold graphs, (complement of) bipartite graphs) and determine their computational complexity. In some of the polynomial-time solvable cases, we also give a structural description of a solution.Graphs and Combinatorics 01/2015; 31(1):73-90. - [Show abstract] [Hide abstract]

**ABSTRACT:**An edge of a k-connected graph G is said to be k-removable if G − e is still k-connected. A subgraph H of a k-connected graph is said to be k-contractible if its contraction, that is, identification every component of H to a single vertex, results again a k-connected graph. In this paper, we show that there is either a removable edge or a contractible subgraph in a 5-connected graph which contains an edge with both endvertices have degree more than five. Thus every edge of minor minimal 5-connected graph is incident to at least one vertex of degree 5.Graphs and Combinatorics 01/2015; 31(1). - [Show abstract] [Hide abstract]

**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). - [Show abstract] [Hide abstract]

**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). - [Show abstract] [Hide abstract]

**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). - [Show abstract] [Hide abstract]

**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). - [Show abstract] [Hide abstract]

**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; -
##### Article: Some Bistar Bipartite Ramsey Numbers

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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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