# Graphs and Combinatorics (GRAPH COMBINATOR)

## Journal 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

## Current impact factor: 0.39

## Impact Factor Rankings

2016 Impact Factor | Available summer 2017 |
---|---|

2014 / 2015 Impact Factor | 0.388 |

2013 Impact Factor | 0.331 |

2012 Impact Factor | 0.351 |

2011 Impact Factor | 0.319 |

2010 Impact Factor | 0.242 |

2009 Impact Factor | 0.571 |

2008 Impact Factor | 0.302 |

2007 Impact Factor | 0.375 |

2006 Impact Factor | 0.175 |

2005 Impact Factor | 0.299 |

2004 Impact Factor | 0.235 |

2003 Impact Factor | 0.159 |

2002 Impact Factor | 0.165 |

2001 Impact Factor | 0.205 |

2000 Impact Factor | 0.085 |

1999 Impact Factor | 0.242 |

1998 Impact Factor | 0.159 |

1997 Impact Factor | 0.183 |

1996 Impact Factor | 0.118 |

1995 Impact Factor | 0.198 |

1994 Impact Factor | 0.215 |

1993 Impact Factor | 0.211 |

1992 Impact Factor | 0.181 |

## Impact factor over time

## Additional details

5-year impact | 0.44 |
---|---|

Cited half-life | 8.30 |

Immediacy index | 0.07 |

Eigenfactor | 0.00 |

Article influence | 0.48 |

Website | Graphs and Combinatorics website |

Other titles | Graphs and combinatorics (Online), Graphs & combinatorics |

ISSN | 0911-0119 |

OCLC | 39980543 |

Material type | Document, Periodical, Internet resource |

Document type | Internet 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

- Classificationgreen

## Publications in this journal

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**ABSTRACT:**In this paper we give a strategy for constructing half-arc-transitive graphs of valency four with prescribed vertex stabilizers. This strategy is applied to answer a 2001 outstanding question of Roman Nedela and Dragan Marušič. - [Show abstract] [Hide abstract]

**ABSTRACT:**A graph G on n vertices is said to be (k, m)-pancyclic if every set of k vertices in G is contained in a cycle of length r for each integer r in the set \(\{ m, m + 1, \ldots , n \}\). This property, which generalizes the notion of a vertex pancyclic graph, was defined by Faudree et al. in (Graphs Combin 20:291–310, 2004). The notion of (k, m)-pancyclicity provides one way to measure the prevalence of cycles in a graph. Broersma and Veldman showed in (Contemporary methods in graph theory, BI-Wiss.-Verlag, Mannheim, Wien, Zürich, pp 181–194, 1990) that any 2-connected claw-free \(P_5\)-free graph must be hamiltonian. In fact, every non-hamiltonian cycle in such a graph is either extendable or very dense. We show that any 2-connected claw-free \(P_5\)-free graph is (k, 3k)-pancyclic for each integer \(k \ge 2\). We also show that such a graph is (1, 5)-pancyclic. Examples are provided which show that these results are best possible. Each example we provide represents an infinite family of graphs. - [Show abstract] [Hide abstract]

**ABSTRACT:**We show that the gap between the two greatest eigenvalues of the generalised Petersen graphs P(n, k) tends to zero as \(n \rightarrow \infty \). Moreover, we provide explicit upper bounds on the size of this gap. It follows that these graphs have poor expansion properties for large values of n. We also show that there is a positive proportion of the eigenvalues of P(n, k) tending to three. - [Show abstract] [Hide abstract]

**ABSTRACT:**Non-reconstructible 3-hypergraphs are studied. A number of counting lemmas are proved for the subgraphs and sub-hypergraphs of a 3-hypergraph. A computer search is used to find all non-reconstructible 3-hypergraphs on at most 8 vertices and 11 triples. A characterization of pseudo-similar vertices in graphs is extended to 3-hypergraphs. - [Show abstract] [Hide abstract]

**ABSTRACT:**Let id(v) denote the implicit degree of a vertex v in a graph G. We define G of order n to be implicit 2-heavy if at least two of the end vertices of each induced claw have implicit degree at least (Formula presented.). In this paper, we show that every implicit 2-heavy graph G is hamiltonian if we impose certain additional conditions on the connectivity of G or forbidden induced subgraphs. Our results extend two previous theorems of Broersma et al. (Discret Math 167–168:155–166, 1997) on the existence of Hamilton cycles in 2-heavy graphs. - [Show abstract] [Hide abstract]

**ABSTRACT:**This paper is primarily concerned with a natural extension of the notion of a total graph \(T(\varGamma (R))\) in the realm of signed graph for a finite commutative ring R. First, we characterize the rings for which the signed total graph \(T_{\varSigma }(\varGamma (R))\) and its line signed graph \(L(T_{\varSigma }(\varGamma (R)))\) are balanced. Second, we characterize the rings for which the negation of a signed total graph \(\eta (T_{\varSigma }(\varGamma (R)))\) is balanced. Several new directions for further research are also indicated. -
##### Article: Clique-Coloring Claw-Free Graphs

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**ABSTRACT:**A clique is defined as a complete subgraph maximal under inclusion and having at least two vertices. A k-clique-coloring of a graph G is an assignment of k colors to the vertices of G such that no clique of G is monochromatic. Bacsó et al. (SIAM J Discrete Math 17:361–376, 2004) noted that the clique-coloring number is unbounded even for the line graphs of complete graphs. In this paper, we prove that a claw-free graph with maximum degree at most 7, except an odd cycle longer than 3, has a 2-clique-coloring by using a decomposition theorem of Chudnovsky and Seymour (J Combin Theory Ser B 98:839–938, 2008) and the limitation of the degree 7 is necessary since the line graph of \(K_{6}\) is a graph with maximum degree 8 but its clique-coloring number is 3 by the Ramsey number \(R(3,3)=6\). In addition, we point out that, if an arbitrary line graph of maximum degree at most d is m-clique-colorable (\(m\ge 3\)), then an arbitrary claw-free graph of maximum degree at most d is also m-clique-colorable. - [Show abstract] [Hide abstract]

**ABSTRACT:**We give a proof of a formula for the Bartholdi zeta function of a digraph by the method of Stark and Terras (Adv. Math. 121:124–165, 1996). - [Show abstract] [Hide abstract]

**ABSTRACT:**We are interested in hereditary classes of graphs (Formula presented.) such that every graph (Formula presented.) satisfies (Formula presented.), where (Formula presented.) ((Formula presented.)) denote the chromatic (clique) number of G. This upper bound is called the Vizing bound for the chromatic number. Apart from perfect graphs few classes are known to satisfy the Vizing bound in the literature. We show that if G is ((Formula presented.), diamond)-free, then (Formula presented.), and we give examples to show that the bound is sharp. - [Show abstract] [Hide abstract]

**ABSTRACT:**Let T be a tournament on n vertices whose arcs are colored with k colors. A 3-cycle whose arcs are colored with three distinct colors is called a rainbow triangle. A rainbow triangle dominated by any nonempty set of vertices is called a dominated rainbow triangle. We prove that when (Formula presented.), if T does not contain a dominated rainbow triangle and all 4- and 5-cycles of T are near-monochromatic, then T has a monochromatic sink. We also prove that when (Formula presented.), if T does not contain a dominated rainbow triangle and all 4-cycles are monochromatic, then T has a monochromatic sink. A semi-cycle is a digraph C that either is a cycle or contains an arc xy such that (Formula presented.) is a cycle. We prove that if (Formula presented.) and all 4-semi-cycles of T are near-monochromatic, then T has a monochromatic sink. We also show if (Formula presented.) and all 5-semi-cycles of T are near-monochromatic, then T has a monochromatic sink. - [Show abstract] [Hide abstract]

**ABSTRACT:**The Kneser graph K(n, k) is the graph whose vertices are the k-element subsets of an n elements set, with two vertices adjacent if they are disjoint. The square \(G^2\) of a graph G is the graph defined on V(G) such that two vertices u and v are adjacent in \(G^2\) if the distance between u and v in G is at most 2. Determining the chromatic number of the square of the Kneser graph K(n, k) is an interesting graph coloring problem, and is also related with intersecting family problem. The square of K(2k, k) is a perfect matching and the square of K(n, k) is the complete graph when \(n \ge 3k-1\). Hence coloring of the square of \(K(2k +1, k)\) has been studied as the first nontrivial case. In this paper, we focus on the question of determining \(\chi (K^2(2k+r,k))\) for \(r \ge 2\). Recently, Kim and Park (Discrete Math 315:69–74, 2014) showed that \(\chi (K^2(2k+1,k)) \le 2k+2\) if \( 2k +1 = 2^t -1\) for some positive integer t. In this paper, we generalize the result by showing that for any integer r with \(1 \le r \le k -2\),(a) \(\chi (K^2 (2k+r, k)) \le (2k+r)^r\), if \(2k + r = 2^t\) for some integer t, and (b) \(\chi (K^2 (2k+r, k)) \le (2k+r+1)^r\), if \(2k + r = 2^t-1\) for some integer t. On the other hand, it was shown in Kim and Park (Discrete Math 315:69–74, 2014) that \(\chi (K^2 (2k+r, k)) \le (r+2)(3k + \frac{3r+3}{2})^r\) for \(2 \le r \le k-2\). We improve these bounds by showing that for any integer r with \(2 \le r \le k -2\), we have \(\chi (K^2 (2k+r, k)) \le 2 \left( \frac{9}{4}k + \frac{9(r+3)}{8} \right) ^r\). Our approach is also related with injective coloring and coloring of Johnson graph. - [Show abstract] [Hide abstract]

**ABSTRACT:**A proper [k]-edge coloring of a graph G is a proper edge coloring of G using colors from \([k]=\{1,2,\ldots ,k\}\). A neighbor sum distinguishing [k]-edge coloring of G is a proper [k]-edge coloring of G such that for each edge \(uv\in E(G)\), the sum of colors taken on the edges incident to u is different from the sum of colors taken on the edges incident to v. By nsdi(G), we denote the smallest value k in such a coloring of G. It was conjectured by Flandrin et al. that if G is a connected graph with at least three vertices and \(G\ne C_5\), then nsdi\((G)\le \varDelta (G)+2\). In this paper, we prove that this conjecture holds for \(K_4\)-minor free graphs, moreover if \(\varDelta (G)\ge 5\), we show that nsdi\((G)\le \varDelta (G)+1\). The bound \(\varDelta (G)+1\) is sharp. - [Show abstract] [Hide abstract]

**ABSTRACT:**A set \(S\subseteq V\) is a paired-dominating set if every vertex in \(V{\setminus } S\) has at least one neighbor in S and the subgraph induced by S contains a perfect matching. The paired-domination number of a graph G, denoted by \(\gamma _{pr}(G)\), is the minimum cardinality of a paired-dominating set of G. A conjecture of Goddard and Henning says that if G is not the Petersen graph and is a connected graph of order n with minimum degree \(\delta (G)\ge 3\), then \(\gamma _{pr}(G)\le 4n/7\). In this paper, we confirm this conjecture for k-regular graphs with \(k\ge 4\). - [Show abstract] [Hide abstract]

**ABSTRACT:**A proper edge-coloring of a graph G is an assignment of colors to the edges of G such that adjacent edges receive distinct colors. A proper edge-coloring defines at each vertex the set of colors of its incident edges. Following the terminology introduced by Horňák, Kalinowski, Meszka and Woźniak, we call such a set of colors the palette of the vertex. What is the minimum number of distinct palettes taken over all proper edge-colorings of G? A complete answer is known for complete graphs and cubic graphs. We study in some detail the problem for 4-regular graphs. In particular, we show that certain values of the palette index imply the existence of an even cycle decomposition of size 3 (a partition of the edge-set of a graph into 3 2-regular subgraphs whose connected components are cycles of even length). This result can be extended to 4d-regular graphs. Moreover, in studying the palette index of a 4-regular graph, the following problem arises: does there exist a 4-regular graph whose even cycle decompositions cannot have size smaller than 4? - [Show abstract] [Hide abstract]

**ABSTRACT:**Let \(K_{n,n}\) be the complete bipartite graph with two parts of equal size n. In this paper, it is shown that depending on whether n is even or odd, \(K_{n,n}\) or \(K_{n,n}-I\), where I is a 1-factor of \(K_{n,n}\), can be decomposed into cycles of distinct even lengths for any integer \(n \ge 2\) with the exception of \(n = 4\). - [Show abstract] [Hide abstract]

**ABSTRACT:**A k-factor of a graph G is a k-regular spanning subgraph of G. A k-factorization is a partition of E(G) into k-factors. Let K(n, p) be the complete multipartite graph with p parts, each of size n. If \(V_{1},\ldots , V_{p}\) are the p parts of V(K(n, p)), then a holey k -factor of deficiency \(V_{i}\) of K(n, p) is a k-factor of \(K(n,p)-V_{i}\) for some i satisfying \(1\le i \le p\). Hence a holey k -factorization is a set of holey k-factors whose edges partition E(K(n, p)). Representing each (holey) k-factor as a color class in an edge-coloring, a (holey) k-factorization of K(n, p) is said to be fair if between each pair of parts the color classes have size within one of each other (so the edges are shared “evenly” among the permitted (holey) factors). In this paper the existence of fair 1-factorizations of K(n, p) is completely settled, as is the existence of fair holey 1-factorizations of K(n, p). The latter result can be used to provide a new construction for symmetric quasigroups of order np with holes of size n. Such quasigroups have the additional property that the permitted symbols are shared as evenly as possible among the cells in each \(n \times n\) “box”. These quasigroups are in some sense as far from frames produced by direct products as possible. - [Show abstract] [Hide abstract]

**ABSTRACT:**A sequence \(a_1a_2\ldots a_p\) is an r-repetition (for a real number \(r >1 \)) if \(p=\lceil rq \rceil \) for some positive integer q, and \(a_j=a_{j+q}\) for \(j=1,2,\ldots , p-q\). In other words, the sequence can be divided into \(\lceil r \rceil \) blocks where all the blocks are the same, say, all the blocks equal to \(a_1a_2\ldots a_q\) for some \(q \ge 1\), except that when r is not an integer, the last block is the prefix of \(a_1...a_q\) of length \( \lceil (r - \lfloor r \rfloor )q \rceil \). A colouring of the vertices of a graph G is r-nonrepetitive if there is no path in G for which the colour sequence of its vertices forms an r-repetition. The r-nonrepetitive chromatic number \(\pi _r(G)\) of G is the minimum number of colours needed in an r-nonrepetitive colouring of G. A k-list assignment of a graph G is a mapping L which assigns a set L(v) of k permissible colours to each vertex v of G. The r-nonrepetitive choice number \(\pi _{rch}(G)\) of G is the least integer k such that for every k-list assignment L, there is an r-nonrepetitive colouring c of G satisfying \(c(v)\in L(v)\) for every vertex v of G. A classical result of Thue asserts that \(\pi _2(P_n)\le 3\) for all n. It is known that \( \pi _{2ch}(P_n) \le 4\) for all n. However, it remains an open problem whether \(\pi _{2ch}(P_n) \le 3\) for all n. This paper proves that for any \(\epsilon > 0\), \(\pi _{(2+\epsilon )ch}(P_n) \le 3\) for all n. - [Show abstract] [Hide abstract]

**ABSTRACT:**Marušič–Scapellato graphs are vertex-transitive graphs of order \(m(2^k + 1)\), where m divides \(2^k - 1\), whose automorphism group contains an imprimitive subgroup that is a quasiprimitive representation of \(\mathrm{SL}(2,2^k)\) of degree \(m(2^k + 1)\). We show that any two Marušič–Scapellato graphs of order pq, where p is a Fermat prime, and q is a prime divisor of \(p - 2\), are isomorphic if and only if they are isomorphic by a natural isomorphism derived from an automorphism of \(\mathrm{SL}(2,2^k)\). This work is a contribution towards the full characterization of vertex-transitive graphs of order a product of two distinct primes.

Data provided are for informational purposes only. Although carefully collected, accuracy cannot be guaranteed. The impact factor represents a rough estimation of the journal's impact factor and does not reflect the actual current impact factor. Publisher conditions are provided by RoMEO. Differing provisions from the publisher's actual policy or licence agreement may be applicable.