Journal Of Graph Theory Research Articles

4-Free Strong Digraphs with the Maximum Size

Directed cycles in digraphs are useful in embedding linear arrays and rings, and are suitable for designing simple algorithm with low communication costs in parallel computer systems, thus the existence of directed cycles on digraphs has been largely investigated. Let [Formula: see text], [Formula: see text] be integers. Bermond et al. [Journal of Graph Theory 4(3) (1980) 337–341] proved that if the size of a strong digraph [Formula: see text] with order [Formula: see text] is at least [Formula: see text], then the girth of [Formula: see text] is no more than [Formula: see text]. Consequently, when [Formula: see text] is a 4-free strong digraph with order [Formula: see text], which means that every directed cycle in [Formula: see text] has length at least [Formula: see text], then the maximum size of [Formula: see text] is [Formula: see text]. In this paper, we mainly give the structural characterizations for all 4-free strong digraphs of order [Formula: see text] whose arc number exactly is [Formula: see text].

Maximal Independent Sets and Maximal Matchings in Series-Parallel and Related Graph Classes

The goal of this paper is to obtain quantitative results on the number and on the size of maximal independent sets and maximal matchings in several block-stable graph classes that satisfy a proper sub-criticality condition. In particular we cover trees, cacti graphs and series-parallel graphs. The proof methods are based on a generating function approach and a proper singularity analysis of solutions of implicit systems of functional equations in several variables. As a byproduct, this method extends previous results of Meir and Moon for trees [Meir, Moon: On maximal independent sets of nodes in trees, Journal of Graph Theory 1988].

Vertex-disjoint cycles in bipartite tournaments

Let k≥2 be an integer. Bermond and Thomassen [Bermond J. C., Thomassen, C., Cycles in digraphs a survey, Journal of Graph Theory 5(1) (1981) 1–43] conjectured that every digraph D with δ+(D)≥2k−1 contains at least k vertex-disjoint cycles. In this work we prove that every bipartite tournament with minimum out-degree at least 2k−2 and minimum in-degree at least one contains k vertex-disjoint cycles of length four, whenever k≥3. Finally, we show that every bipartite tournament with minimum degree at least (3k−1)/2 contains k vertex-disjoint cycles of length four.

Induced minors and well-quasi-ordering

A graph H is an induced minor of a graph G if it can be obtained from an induced subgraph of G by contracting edges. Otherwise, G is said to be H-induced minor-free. Robin Thomas showed in [Graphs without K4and well-quasi-ordering, Journal of Combinatorial Theory, Series B, 38(3):240 – 247, 1985] that K4-induced minor-free graphs are well-quasi ordered by induced minors.We provide a dichotomy theorem for H-induced minor-free graphs and show that the class of H-induced minor-free graphs is well-quasi-ordered by the induced minor relation if and only if H is an induced minor of the gem (the path on 4 vertices plus a dominating vertex) or of the graph obtained by adding a vertex of degree 2 to the complete graph on 4 vertices.Similar dichotomy results were previously given by Guoli Ding in [Subgraphs and well-quasi-ordering, Journal of Graph Theory, 16(5):489–502, 1992] for subgraphs and Peter Damaschke in [Induced subgraphs and well-quasi-ordering, Journal of Graph Theory, 14(4):427–435, 1990] for induced subgraphs.

Antimagic Labeling of Regular Graphs

A graph $G=(V,E)$ is antimagic if there is a one-to-one correspondence $f: E \to \{1,2,\ldots, |E|\}$ such that for any two vertices $u,v$, $\sum_{e \in E(u)}f(e) \ne \sum_{e\in E(v)}f(e)$. It is known that bipartite regular graphs are antimagic and non-bipartite regular graphs of odd degree at least three are antimagic. Whether all non-bipartite regular graphs of even degree are antimagic remained an open problem. In this paper, we solve this problem and prove that all even degree regular graphs are antimagic. The paper was submitted to December 2014 by Journal of Graph Theory.

Rainbow Connection Number and Connected Dominating Sets

Abstract Rainbow connection number, rc(G), of a connected graph G is the minimum number of colours needed to colour the edges of G, so that every pair of vertices is connected by at least one path in which no two edges are coloured the same. In this paper we show that for every connected graph G, with minimum degree at least 2, r c ( G ) ⩽ γ c ( G ) + 2 , where γ c ( G ) is the connected domination number of G. Bounds of the form d i a m e t e r ( G ) ⩽ r c ( G ) ⩽ d i a m e t e r ( G ) + c , 1 ⩽ c ⩽ 4 , for many special graph classes follow as easy corollaries from this result. We also show that every bridge-less chordal graph G has r c ( G ) ⩽ 3 ⋅ r a d i u s ( G ) . An extension of this idea to two-step dominating sets is used to show that for every connected graph on n vertices with minimum degree δ , r c ( G ) ⩽ 3 n / ( δ + 1 ) + 3 . This solves an open problem from Schiermeyer, 2009 [Schiermeyer, I. (2009). Rainbow connection in graphs with minimum degree three. In Fiala, J., Kratochvl, J., and Miller, M., editors, Combinatorial Algorithms, volume 5874 of Lecture Notes in Computer Science, pages 432–437. Springer Berlin / Heidelberg.], improving the previously best known bound of 20 n / δ from Krivelevich and Yuster, 2010 [Krivelevich, M. and Yuster, R. (2010). The rainbow connection of a graph is (at most) reciprocal to its minimum degree. Journal of Graph Theory, 63(3):185–191.]. Moreover, this bound is seen to be tight up to additive factors by a construction mentioned in Caro et. al., 2008 [Caro, Y., Lev, A., Roditty, Y., Tuza, Z., and Yuster, R. (2008). On rainbow connection. The electronic journal of combinatorics, 15(R57):1].

Contributors

Contributors

Edge-connectivity of permutation hypergraphs

In this note we provide a generalization of a result of Goddard et al. (2003) [4] on edge-connectivity of permutation graphs for hypergraphs. A permutation hypergraph Gπ is obtained from a hypergraph G by taking two disjoint copies of G and by adding a perfect matching between them. The main tool in the proof of the graph result was the theorem on partition constrained splitting off preserving k-edge-connectivity due to Bang-Jensen et al. (1999) [1]. Recently, this splitting off theorem was extended for hypergraphs by Bernáth et al. (accepted in Journal of Graph Theory) [2]. This extension made it possible to find a characterization of hypergraphs for which there exists a k-edge-connected permutation hypergraph.

Bounding [formula omitted] in terms of [formula omitted] and [formula omitted] for some classes of graphs

Reed [B. Reed, ω , Δ and χ , Journal of Graph Theory, 27 (1998) 177–212] conjectured that for any graph G , χ ( G ) ≤ ⌈ Δ ( G ) + ω ( G ) + 1 2 ⌉ , where χ ( G ) , ω ( G ) , and Δ ( G ) respectively denote the chromatic number, the clique number and the maximum degree of G . In this paper, we verify this conjecture for some special classes of graphs which are defined by families of forbidden induced subgraphs.

Yueh-Gin Gung and Dr. Charles Y. Hu Award for 2011 to Joseph A. Gallian for Distinguished Service to Mathematics

The two themes that run through Joseph A. Gallian’s service to mathematics are (a) encouraging young mathematicians and helping them to develop successful careers and (b) communicating mathematics to the widest possible audience. He was one of the early proponents of undergraduates conducting mathematical research, and his REU at Duluth, which began in 1977, is widely regarded as the premier REU. The quality of the work at Joe’s REU is evidenced by the 160 papers by participants that grew out of their REU work. These papers have appeared in such journals as Crelle’s Journal, Journal of Algebra, Journal of Combinatorial Theory, Discrete Mathematics, Applied Discrete Mathematics, Annals of Discrete Mathematics, and Journal of Graph Theory. The REU, along with Joe’s continuing contact with its participants, makes an important contribution to developing the next generations of mathematicians. Participants have included some prominent mathematicians, whose careers this REU helped to shape. Joe is also an inspiration to a generation of mathematicians who involve students in high-quality undergraduate research in mathematics. Not only is Joe successful with his own REU, but he is generous with his time and advice to help others to set up REUs. In 2002, Joe was recognized by the Council on Undergraduate Research with their Fellow Award, given to members who have demonstrated sustained excellence in research with undergraduates.

Each 3-strong Tournament Contains 3 Vertices Whose Out-arcs Are Pancyclic

An arc in a tournament T with n ≥ 3 vertices is called pancyclic, if it is in a cycle of length k for all 3 ≤ k ≤ n. Yeo (Journal of Graph Theory, 50 (2005), 212–219) proved that every 3-strong tournament contains two distinct vertices whose all out-arcs are pancyclic, and conjectured that each 2-strong tournament contains 3 such vertices. In this paper, we confirm Yeo’s conjecture for 3-strong tournaments.

Planar Graphs without 7-Cycles Are 4-Choosable

Previous article Next article Planar Graphs without 7-Cycles Are 4-ChoosableBabak FarzadBabak Farzadhttps://doi.org/10.1137/05064477XPDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractIn this paper, we show that every planar graph without 7-cycles is 4-choosable.[1] S. Choudum, Some 4-valent, 3-connected, planar, almost pancyclic graphs, Discrete Math., 18 (1977), pp. 125–129. DSMHA4 0012-365X CrossrefISIGoogle Scholar[2] G. Fijavz, , M. Juvan, , B. Mohar and , and R. Skrekovski, Planar graphs without cycles of specific lengths, European J. Combin., 23 (2002), pp. 377–388. EJOCDI 0195-6698 CrossrefISIGoogle Scholar[3] S. Gutner, The complexity of planar graph choosability, Discrete Math., 159 (1996), pp. 119–130. DSMHA4 0012-365X CrossrefISIGoogle Scholar[4] P. Lam, , W. Shiu and , and B. Xu, On structure of some plane graphs with application to choosability, J. Combin. Theory Ser. B, 82 (2001), pp. 285–296. JCBTB8 0095-8956 CrossrefISIGoogle Scholar[5] P. Lam, , B. Xu and , and J. Liu, The 4-choosability of plane graphs without 4-cycles, J. Combin. Theory Ser. B, 76 (1999), pp. 117–126. JCBTB8 0095-8956 CrossrefISIGoogle Scholar[6] C. Thomassen, Every planar graph is 5-choosable, J. Combin. Theory Ser. B, 62 (1994), pp. 180–181. JCBTB8 0095-8956 CrossrefISIGoogle Scholar[7] M. Voigt, List colourings of planar graphs, Discrete Math., 120 (1993), pp. 215–219. DSMHA4 0012-365X CrossrefISIGoogle Scholar[8] M. Voigt and and B. Wirth, On 3-colorable non-4-choosable planar graphs, J. Graph Theory, 24 (1997), pp. 233–235. JGTHDO 0364-9024 CrossrefISIGoogle Scholar[9] W. F. Wang and and K. W. Lih, The 4-choosability of planar graphs without 6-cycles, Australas. J. Combin., 24 (2001), pp. 157–164. 1034-4942 Google Scholar[10] W. F. Wang and and K. W. Lih, Choosability and edge choosability of planar graphs without five cycles, Appl. Math. Lett., 15 (2002), pp. 561–565. AMLEEL 0893-9659 CrossrefISIGoogle ScholarKeywordsplanar graphscoloringchoosability Previous article Next article FiguresRelatedReferencesCited byDetails A sufficient condition for planar graphs to be DP -4-colorable7 October 2022 | Quaestiones Mathematicae, Vol. 5 Cross Ref Two sufficient conditions for a planar graph to be list vertex-2-arborableDiscrete Mathematics, Vol. 345, No. 6 Cross Ref Cover and variable degeneracyDiscrete Mathematics, Vol. 345, No. 4 Cross Ref DP-4-coloring of planar graphs with some restrictions on cyclesDiscrete Mathematics, Vol. 344, No. 11 Cross Ref DP-coloring on planar graphs without given adjacent short cycles10 October 2020 | Discrete Mathematics, Algorithms and Applications, Vol. 13, No. 02 Cross Ref Planar Graphs Without Pairwise Adjacent $3$-, $4$-, $5$-, and $6$-cycle are $4$-choosableTaiwanese Journal of Mathematics, Vol. -1, No. -1 Cross Ref Planar graphs without 7-cycles and butterflies are DP-4-colorableDiscrete Mathematics, Vol. 343, No. 8 Cross Ref DP-4-colorability of planar graphs without adjacent cycles of given lengthDiscrete Applied Mathematics, Vol. 277 Cross Ref List coloring and diagonal coloring for plane graphs of diameter twoApplied Mathematics and Computation, Vol. 363 Cross Ref Facial Colorings of Plane Graphs28 April 2019 | Journal of Interconnection Networks, Vol. 19, No. 01 Cross Ref Planar graphs without chordal 6-cycles are 4-choosableDiscrete Applied Mathematics, Vol. 244 Cross Ref Planar graphs without intersecting 5 -cycles are 4 -choosableDiscrete Mathematics, Vol. 340, No. 8 Cross Ref (4, 2)-Choosability of Planar Graphs with Forbidden Structures14 June 2017 | Graphs and Combinatorics, Vol. 33, No. 4 Cross Ref A sufficient condition for a planar graph to be 4 -choosableDiscrete Applied Mathematics, Vol. 224 Cross Ref Toroidal graphs containing neither K5− nor 6-cycles are 4-choosable26 May 2016 | Journal of Graph Theory, Vol. 85, No. 1 Cross Ref Planar graphs without 4-cycles adjacent to triangles are 4-choosableDiscrete Mathematics, Vol. 339, No. 12 Cross Ref A Note on 3-choosability of Planar Graphs Related to Montanssier’s Conjecture20 November 2018 | Canadian Mathematical Bulletin, Vol. 59, No. 2 Cross Ref Improper Choosability of Planar Graphs without 4-CyclesYingqian Wang and Lingji Xu5 December 2013 | SIAM Journal on Discrete Mathematics, Vol. 27, No. 4AbstractPDF (192 KB)On Coloring Problems26 July 2013 Cross Ref On the vertex-arboricity of planar graphs without 7-cyclesDiscrete Mathematics, Vol. 312, No. 15 Cross Ref Choosability of toroidal graphs without short cyclesJournal of Graph Theory, Vol. 28 Cross Ref Volume 23, Issue 3| 2009SIAM Journal on Discrete Mathematics History Submitted:29 November 2005Accepted:06 April 2009Published online:15 July 2009 InformationCopyright © 2009 Society for Industrial and Applied MathematicsKeywordsplanar graphscoloringchoosabilityMSC codes05C1505C10PDF Download Article & Publication DataArticle DOI:10.1137/05064477XArticle page range:pp. 1179-1199ISSN (print):0895-4801ISSN (online):1095-7146Publisher:Society for Industrial and Applied Mathematics

Another proof of Wright's inequalities

We present a short way of proving the inequalities obtained by Wright in [E.M. Wright, The number of connected sparsely edged graphs III: Asymptotic results, Journal of Graph Theory 4 (1980) 393–407] concerning the number of connected graphs with ℓ edges more than vertices.

Erratum to: Factorizations and characterizations of induced‐hereditary and compositive properties

AbstractThe original article to which this Erratum refers was published in Journal of Graph Theory 49:11–27.No Abstract.

Factorizations and characterizations of induced‐hereditary and compositive properties

AbstractAn Erratum has been published for this article in Journal of Graph Theory 50:261, 2005.A graph property (i.e., a set of graphs) is hereditary (respectively, induced‐hereditary) if it is closed under taking subgraphs (resp., induced‐subgraphs), while the property is additive if it is closed under disjoint unions. If ${\cal P}$ and ${\cal Q}$ are properties, the product ${\cal P}\circ {\cal Q}$ consists of all graphs G for which there is a partition of the vertex set of G into (possibly empty) subsets A and B with G[A] $\in {\cal P}$ and G[B] $\in {\cal Q}$. A property is reducible if it is the product of two other properties, and irreducible otherwise.We show that very few reducible induced‐hereditary properties have a unique factorization into irreducibles, and we describe them completely. On the other hand, we give a new and simpler proof that additive hereditary properties have a unique factorization into irreducible additive hereditary properties [J. Graph Theory 33 (2000), 44–53]. We also introduce analogs of additive induced‐hereditary properties, and characterize them in the style of Scheinerman [Discrete Math. 55 (1985), 185–193]. © 2005 Wiley Periodicals, Inc. J Graph Theory 49: 11–27, 2005

D. Liu and X. Zhu, Erratum to: “Fractional chromatic number and circular chromatic number for distance graphs with large clique size”. Journal of Graph Theory47(2) 2004, 129–146

AbstractThe original article to which this Erratum refers was published in Journal of Graph Theory 47:129–146,2004.

On Clique-inverse graphs of Kp-free graphs

Abstract In this paper, for p ≥ 4 , we prove that the family of forbidden induced subgraphs for a graph G such that the clique graph of G is Kp-free is finite, providing a proof of a conjecture in [F. Protti and J. Szwarcfiter. Clique-Inverse Graphs of K3-free and K4-free Graphs, Journal of Graph Theory (2000) 257–272]. We also give an upper bound for the rank of Sperner conformal hypergraphs that contain p pairwise intersecting edges and are minimal with respect to all these properties.

Fractional chromatic number and circular chromatic number for distance graphs with large clique size

An Erratum has been published for this article in Journal of Graph Theory 48: 329330, 2005. Let M be a set of positive integers. The distance graph generated by M, denoted by G(Z, M), has the set Z...

Cycles of given color patterns

In 2-edge-colored graphs, we define an (s, t)-cycle to be a cyle of length s + t, in which s consecutive edges are in one color and the remaining t edges are in the other color. Here we investigate the existence of (s, t)-cycles, in a 2-edge-colored complete graph Kcn on n vertices. In particular, in the first result we give a complete characterization for the existence of (s, t)-cycles in Kcn with n relatively large with respect to max({s, t}). We also study cycles of length 4 for all possible values of s and t. Then, we show that Kcn contains an (s, t)-hamiltonian cycle unless it is isomorphic to a specified graph. This extends a result of A. Gyárfás [Journal of Graph Theory, 7 (1983), 131–135]. Finally, we give some sufficient conditions for the existence of (s, 1)-cycles, (inverted sans serif aye) s ϵ {2, 3,…, n − 2}. © 1996 John Wiley & Sons, Inc.

Cycles of given color patterns

In 2-edge-colored graphs, we define an (s, t)-cycle to be a cyle of length s + t, in which s consecutive edges are in one color and the remaining t edges are in the other color. Here we investigate the existence of (s, t)-cycles, in a 2-edge-colored complete graph Kcn on n vertices. In particular, in the first result we give a complete characterization for the existence of (s, t)-cycles in Kcn with n relatively large with respect to max({s, t}). We also study cycles of length 4 for all possible values of s and t. Then, we show that Kcn contains an (s, t)-hamiltonian cycle unless it is isomorphic to a specified graph. This extends a result of A. Gyarfas [Journal of Graph Theory,7 (1983), 131–135]. Finally, we give some sufficient conditions for the existence of (s, 1)-cycles, (inverted sans serif aye) s ϵ {2, 3,…, n − 2}. © 1996 John Wiley & Sons, Inc.