Comb Theory Research Articles

A Twelve Vertex Theorem for 3-Connected Claw-Free Graphs

The cyclability of a graph H, denoted by C(H), is the largest integer r such that H has a cycle through any r vertices. For a claw-free graph H, by Ryjaă?ek (J Comb Theory Ser B 70:217---224, 1997) closure concept, there is a $$K_3$$K3-free graph G such that the closure $$cl(H)=L(G)$$cl(H)=L(G). In this note, we prove that for a 3-connected claw-free graph H with its closure $$cl(H)=L(G)$$cl(H)=L(G), $$C(H)\ge 12$$C(H)?12 if and only if G can not be contracted to the Petersen graph in such a way that each vertex in P is obtained by contracting a nontrivial connected $$K_3$$K3-free subgraph. This is an improvement of the main result in Gyori and Plummer (Stud Sci Math Hung 38:233---244, 2001).

Tree Connectivities of Cayley Graphs on Abelian Groups with Small Degrees

The generalized k-connectivity $$\kappa _{k}(G)$$ and the generalized k-edge-connectivity $$\lambda _k(G)$$ of a graph G, also known as the tree connectivities, were introduced by Hager (J Comb Theory Ser B 38:179–189, 1985) and Li et al. (Discret Math Theor Comput Sci 14:43–54, 2012), respectively. In this paper, we study these invariants for Cayley graphs on Abelian groups with degree 3 or 4. When G is cubic, we prove $$\kappa _k(G) = \lambda _k(G) = 2$$ for $$3\le k\le 6$$ and $$\kappa _k(G) = \lambda _k(G) = 1$$ for $$7\le k\le n$$ . When G has degree 4, we obtain $$\kappa _{3}(G)=\lambda _3(G)=3$$ , $$\lambda _k(G) = 2$$ and $$\kappa _k(G)\le 2$$ for $$k\ge 8$$ , and $$\kappa _k(G) = 2$$ for $$k=n-1,n$$ .

A construction of weakly and non-weakly regular bent functions over the ring of integers modulo $$p^m$$ p m

Bent functions over the finite fields of an odd characteristic received a lot of attention of late years. In Cesmelioglu et al. (J Comb Theory Ser A 119:420---429, 2012), Cesmelioglu and Meidl (Des Codes Cryptogr 66:231---242, 2013), an efficient method of construction of weakly regular and non-weakly regular bent functions defined over a finite field with odd characteristic is presented. In this paper, we give an adaptation of this method to the ring of integers modulo $$p^m$$pm, where p is an odd prime and m is a positive integer. We emphasize that different results than the results of the finite field case are obtained in every application process. First, we give a method that constructs bent functions using plateaued functions by increasing the dimension. Then, in order to give concrete examples, we compute Walsh spectrum of some specific quadratic functions defined over the ring of integers modulo $$p^m$$pm and apply the construction method on these functions. There are notable differences between the cases when m is odd and even. Also, we explain how to determine weakly regular and non-weakly regular bent functions among the bent functions that are constructed by the method.

Finding Shorter Cycles in a Weighted Graph

In this paper we investigate the structures of short cycles in a weighted graph. By Thomassen's $$3$$3-path-condition theory (Thomassen in J Comb Theory Ser 48:155---177, 1990), it is easy to find a shortest cycle in a collection of cycles beyond a subspace of the cycle space of a graph. What is more difficult for one to do is to find a shortest cycle within a subspace of the cycle space in polynomial time. By using the Dijkstra's algorithm (Dijkstra in Numer Math 1:55---61, 1959) we find a collection $$\mathcal {C}$$C of cycles containing many types of short cycles within a given subspace of the cycle space of a graph and this implies a polynomial time algorithm (called extended fundamental cycle algorithm) to locate all the possible shortest cycles in a weighted graph. In the case of unweighted graphs, the algorithm may also find every shortest even cycle in a graph, this greatly improved a result of Grotschel and Pulleyblank (Oper Res Lett 1:23---27, 1981/82), Monien (Computing 31:355---369, 1983), Yuster and Zwick (SIAM J Discrete Math 10:209---222, 1997). In fact, our algorithm shows that there are at most $$O(n^4)$$O(n4) many such short cycles in an unweighted graph of order $$n$$n. Further more, our fundamental cycle method may find a minimum cycle base (or simply MCB as some scholars named) in the cycle space of a graph. Since the structure of MCB's is unique (Ren and Deng in Discrete Math 307:2654---2660, 2007), this shows that, in the sense, cycles in a MCB are nearly-fundamental (i.e., each element in a MCB is a sum of at most two fundamental cycles). This provides a new way to study MCB.

On a Coloring Conjecture of Hajós

Hajos conjectured that graphs containing no subdivision of $$K_5$$K5 are 4-colorable. It is shown in Yu and Zickfeld (J Comb Theory Ser B 96:482---492, 2006) that if there is a counterexample to this conjecture then any minimum such counterexample must be 4-connected. In this paper, we further show that if $$G$$G is a minimum counterexample to Hajos' conjecture and $$S$$S is a 4-cut in $$G$$G then $$G-S$$G-S has exactly two components.

A classification of semisymmetric graphs of order $$2p^3$$ 2 p 3 : unfaithful case

A simple undirected graph is said to be semisymmetric if it is regular and edge-transitive, but not vertex-transitive. Every semisymmetric graph is a bipartite graph with two biparts of equal size. It was proved by Folkman in (J Comb Theory Ser B 3:215–232, 1967) that there exist no semisymmetric graphs of order $$2p$$ and $$2p^2$$ , where $$p$$ is a prime. For any distinct primes $$p$$ and $$q$$ , the classification of semisymmetric graphs of order $$2pq$$ was given by Du and Xu in (Comm Algebra 28:2685–2715, 2000). Naturally, one of our long-term goals is to determine all the semisymmetric graphs of order $$2p^3$$ , for any prime $$p$$ . All these graphs $$\Gamma $$ are divided into two subclasses: (I) the automorphism group $$\hbox {Aut }(\Gamma )$$ acts unfaithfully on at least one bipart; and (II) $$\hbox {Aut }(\Gamma )$$ acts faithfully on both biparts. In Wang and Du (Eur J Comb 36:393–405, 2014), a group theoretical characterization for Subclass (I) was given by the authors. Based on this characterization, this paper gives a complete classification for Subclass (I).

List coloring triangle-free hypergraphs

A triangle in a hypergraph is a collection of distinct vertices u, v, w and distinct edges e, f, g with , and . Johansson [Tech. report (1996)] proved that every triangle-free graph with maximum degree Δ has list chromatic number . Frieze and Mubayi (Electron J Comb 15 (2008), 27) proved that every linear (meaning that every two edges share at most one vertex) triangle-free triple system with maximum degree Δ has chromatic number . The restriction to linear triple systems was crucial to their proof. We provide a common generalization of both these results for rank 3 hypergraphs (edges have size 2 or 3). Our result removes the linear restriction from [8], while reducing to the (best possible) result [Johansson, Tech. report (1996)] for graphs. In addition, our result provides a positive answer to a restricted version of a question of Ajtai Erdős, Komlos, and Szemeredi (combinatorica 1 (1981), 313–317) concerning sparse 3-uniform hypergraphs. As an application, we prove that if is the collection of 3-uniform triangles, then the Ramsey number satisfies for some positive constants a and b. The upper bound makes progress towards the recent conjecture of Kostochka, Mubayi, and Verstraete (J Comb Theory Ser A 120 (2013), 1491–1507) that where C3 is the linear triangle. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 487–519, 2015

Nonsymmetric primitive translation schemes of prime power order

It is well-known that translation schemes of prime order are exactly the cyclotomic schemes. In this paper, we show that there do not exist any two-, three-, or four-class nonsymmetric primitive translation schemes of prime square order. On the other hand, we find new nonsymmetric four- and five-class association schemes from cyclotomy as fission schemes of certain symmetric three-class association schemes. Moreover, we provide an affirmative answer to the following question raised by Song (J Algebr Comb 5:47---55, 1996): are there any other two-class primitive association schemes that admit symmetrizable fission schemes besides the cyclotomic scheme of index 2 for $$q \equiv 5 \hbox { (mod 8)}$$ q � 5 (mod 8) ? To be more specific, we show that a certain two-class primitive association scheme in the finite field $${\mathbb F}_{37^3}$$ F 37 3 constructed by Feng and Xiang (J Comb Theory Ser A 119:245---256, 2012) admits a four-class fission scheme. This fission scheme is realized as a fusion scheme of the cyclotomic scheme of index 28.

Group connectivity in $$J_3$$ J 3 line graphs

Jaeger et al. (J Comb Theory Ser B 56:165---182, 1992) conjectured that every 5-edge-connected graph is $$Z_3$$Z3-connected. Moreover, Lai et al. (Discret Math 311:2295---2307, 2011) proved that every 5-edge-connected graph is $$Z_3$$Z3-connected if and only if every 5-edge-connected line graph is $$Z_3$$Z3-connected. A graph $$G$$G is a $$J_3$$J3 graph if every edge of $$G$$G lies in a 3-cycle of $$G$$G. We prove that every 5-edge-connected $$J_3$$J3 line graph is $$Z_3$$Z3-connected.

On Panchromatic Digraphs and the Panchromatic Number

Let D = (V, A) be a simple finite digraph, and let $${\pi(D)}$$ ? ( D ) , the panchromatic number of D, be the maximum number of colours k such that for each (effective, or onto) colouring of its arcs $${\varsigma : A \to [k]}$$ ? : A ? [ k ] a monochromatic path kernel N ? V, as introduced in Galeana-Sanchez (Discrete Math 184: 87---99, 1998), exists. It is not hard to see that D has a kernel--in the sense of Von Neumann and Morgenstern (Theory of games and economic behaviour. Princeton University Press, Princeton, 1944)--if and only if $${\pi(D) = |A|}$$ ? ( D ) = | A | . In this note this invariant is introduced and some of its structural bounds are studied. For example, the celebrated theorem of Sands et al. (J Comb Theory Ser 33: 271---275, 1982), in terms of this invariant, settles that $${\pi(D) \geq 2}$$ ? ( D ) ? 2 . It will be proved that $$\pi(D) < |A| \iff \pi(D) < \min \left\{2\sqrt{\chi(D)}, \chi(L(D)), \theta(D) + \max {\text d}_c(K_i) + 1\right\},$$ ? ( D ) < | A | ? ? ( D ) < min 2 ? ( D ) , ? ( L ( D ) ) , ? ( D ) + max d c ( K i ) + 1 , where $${\chi(\cdot)}$$ ? ( · ) denotes the chromatic number, $${L(\cdot)}$$ L ( · ) denotes the line digraph, $${\theta(\cdot)}$$ ? ( · ) denotes the minimum partition into complete graphs of the underlying graph and $${\text{d}_c(\cdot)}$$ d c ( · ) denotes the dichromatic number. We also introduce the notion of a panchromatic digraph which is a digraph D such that for every k ≤ |A| and every k-colouring of its arcs, it has a monochromatic path kernel. Some classes of panchromatic digraphs are further characterised.

Right-convergence of sparse random graphs

The paper is devoted to the problem of establishing right-convergence of sparse random graphs. This concerns the convergence of the logarithm of number of homomorphisms from graphs or hyper-graphs \(\mathbb{G }_N, N\ge 1\) to some target graph \(W\). The theory of dense graph convergence, including random dense graphs, is now well understood (Borgs et al. in Ann Math 176:151–219, 2012; Borgs et al. in Adv Math 219:1801–1851, 2008; Chatterjee and Varadhan in Eur J Comb 32:1000–1017, 2011; Lovász and Szegedy in J Comb Theory Ser B 96:933–957, 2006), but its counterpart for sparse random graphs presents some fundamental difficulties. Phrased in the statistical physics terminology, the issue is the existence of the limits of appropriately normalized log-partition functions, also known as free energy limits, for the Gibbs distribution associated with \(W\). In this paper we prove that the sequence of sparse Erdös-Rényi graphs is right-converging when the tensor product associated with the target graph \(W\) satisfies a certain convexity property. We treat the case of discrete and continuous target graphs \(W\). The latter case allows us to prove a special case of Talagrand’s recent conjecture [more accurately stated as level III Research Problem 6.7.2 in his recent book (Talagrand in Mean Field Models for Spin Glasses: Volume I: Basic examples. Springer, Berlin, 2010)], concerning the existence of the limit of the measure of a set obtained from \(\mathbb{R }^N\) by intersecting it with linearly in \(N\) many subsets, generated according to some common probability law. Our proof is based on the interpolation technique, introduced first by Guerra and Toninelli (Commun Math Phys 230:71–79, 2002) and developed further in (Abbe and Montanari in On the concentration of the number of solutions of random satisfiability formulas, 2013; Bayati et al. in Ann Probab Conference version in Proceedings of 42nd Ann. Symposium on the Theory of Computing (STOC), 2010; Contucci et al. in Antiferromagnetic Potts model on the Erdös-Rényi random graph, 2011; Franz and Leone in J Stat Phys 111(3/4):535–564, 2003; Franz et al. in J Phys A Math Gen 36:10967–10985, 2003; Montanari in IEEE Trans Inf Theory 51(9):3221–3246, 2005; Panchenko and Talagrand in Probab Theory Relat Fields 130:312–336, 2004). Specifically, Bayati et al. (Ann Probab Conference version in Proceedings of 42nd Ann. Symposium on the Theory of Computing (STOC), 2010) establishes the right-convergence property for Erdös-Rényi graphs for some special cases of \(W\). In this paper most of the results in Bayati et al. (Ann Probab Conference version in Proceedings of 42nd Ann. Symposium on the Theory of Computing (STOC), 2010) follow as a special case of our main theorem.

Group divisible designs with block size four and group type $$g^um^1$$ g u m 1

Non-uniform group divisible designs have been studied by numerous researchers in the past two decades due to their vital applications in the constructions for other types of designs. Much progress has been made for the existence of $$\{4\}$$ { 4 } -GDDs of type $$g^um^1$$ g u m 1 , especially when $$gu$$ g u is even. The corresponding problem for block size three had been solved by Colbourn et al. (J Comb Theory Ser A 59:73---89, 1992). In this paper, we consider the entire existence problem for such $$\{4\}$$ { 4 } -GDDs. We show that, for each given g, up to a small number of undetermined cases of $$u$$ u , the necessary conditions on $$(u,m)$$ ( u , m ) for the existence of a $$\{4\}$$ { 4 } -GDD of type $$g^um^1$$ g u m 1 are also sufficient.

Classifying Cubic Semisymmetric Graphs of Order 18 p n

A graph X is said to be G-semisymmetric if it is regular and there exists a subgroup G of A := Aut (X) acting transitively on its edge set but not on its vertex set. In the case of G = A, we call X a semisymmetric graph. Let p be a prime. It was shown by Folkman (J Comb Theory 3:215–232, 1967) that a regular edge-transitive graph of order 2p or 2p 2 is necessarily vertex-transitive. The smallest semisymmetric graph is the Folkman graph. In this study, we classify all connected cubic semisymmetric graphs of order 18p n , where p is a prime and $${n \geq 1}$$ .

Metric intersection problems in Cayley graphs and the Stirling recursion

In Sym(n) with n ≥ 5 let H be a conjugacy class of elements of order 2 and let Γ be the Cayley graph whose vertex set is the group G generated by H (so G = Sym(n) or Alt(n)) and whose edge set is determined by H. We are interested in the metric structure of this graph. In particular, for $${g \in G}$$ let B r (g) be the metric ball in Γ of radius r and centre g. We show that the intersection numbers $${\Phi(\Gamma; r, g) := |\,B_{r}(e)\,\cap\,B_{r}(g)\,|}$$ are generalized Stirling functions in n and r. The results are motivated by the study of error graphs in Levenshtein (Dokl Akad Nauk 354:593–596, 1997; IEEE Trans Inform Theory 47(1):2–22, 2001; (J Comb Theory Ser A 93(2):310–332, 2001) and related reconstruction problems.

Quadratic Upper Bounds on the Erdős–Pósa Property for a Generalization of Packing and Covering Cycles

AbstractAccording to the classical Erdős–Pósa theorem, given a positive integer k, every graph G either contains k vertex disjoint cycles or a set of at most vertices that hits all its cycles. Robertson and Seymour (J Comb Theory Ser B 41 (1986), 92–114) generalized this result in the best possible way. More specifically, they showed that if is the class of all graphs that can be contracted to a fixed planar graph H, then every graph G either contains a set of k vertex‐disjoint subgraphs of G, such that each of these subgraphs is isomorphic to some graph in or there exists a set S of at most vertices such that contains no subgraph isomorphic to any graph in . However, the function f is exponential. In this note, we prove that this function becomes quadratic when consists all graphs that can be contracted to a fixed planar graph . For a fixed c, is the graph with two vertices and parallel edges. Observe that for this corresponds to the classical Erdős–Pósa theorem.

On independent sets in hypergraphs

AbstractThe independence number of a hypergraph H is the size of a largest set of vertices containing no edge of H. In this paper, we prove that if Hn is an n‐vertex ‐uniform hypergraph in which every r‐element set is contained in at most d edges, where , then urn:x-wiley::media:rsa20453:rsa20453-math-0004 where satisfies as . The value of cr improves and generalizes several earlier results that all use a theorem of Ajtai, Komlós, Pintz, Spencer and Szemerédi (J Comb Theory Ser A 32 (1982), 321–335). Our relatively short proof extends a method due to Shearer (Random Struct Algorithms 7 (1995), 269–271) and Alon (Random Struct Algorithms 9 (1996), 271–278). The above statement is close to best possible, in the sense that for each and all values of , there are infinitely many Hn such that urn:x-wiley::media:rsa20453:rsa20453-math-0010 where depends only on r. In addition, for many values of d we show as , so the result is almost sharp for large r. We give an application to hypergraph Ramsey numbers involving independent neighborhoods.Copyright © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 224‐239, 2014

Two Equivalent Properties of $$\mathcal{Z}_3$$ -Connectivity

The concept of group connectivity was introduced by Jaeger et al. (J Comb Theory Ser B 56:165---182, 1992) for the study of integer flows. The concept of all generalized Tutte-orientations was introduced by Barat and Thomassen (J Graph Theory 52:135---146, 2006) for the study of claw-decompositions of graphs. In this paper, we establish the equivalence of the following 3 properties: a graph is $$\mathcal{Z}_3$$ -connected, a graph admits all generalized Tutte-orientations and a graph is 3-flow contractible. We also give some applications of this result.

Small tight sets of hyperbolic quadrics

An x-tight set of a hyperbolic quadric Q+(2n + 1, q) can be described as a set M of points with the property that the number of points of M in the tangent hyperplanes of points of M is as big as possible. We show that such a set is necessarily the union of x mutually disjoint generators provided that x ≤ q and n ≤ 3, or that x < q, n ≥ 4 and q ≥ 71. This unifies and generalizes many results on x-tight sets that are presently known, see (J Comb Theory Ser A 114(7):1293–1314 [1], J Comb Des 16(4):342–349 [5], Des Codes Cryptogr 50:187–201 [4], Adv Geom 4(3):279–286 [8], Bull Lond Math Soc 42(6):991–996 [11]).

A Characterization of the Locally Finite Networks Admitting Non-Constant Harmonic Functions of Finite Energy

We characterize the locally finite networks admitting non-constant harmonic functions of finite energy. Our characterization unifies the necessary existence criteria of Thomassen (J Comb Theory, Ser B 49:87–102, 1990) and of Lyons and Peres (2011) with the sufficient criterion of Soardi (1991). We also extend a necessary existence criterion for non-elusive non-constant harmonic functions of finite energy due to Georgakopoulos (J Lond Math Soc, 2010).

Singer 8-arcs of Mathon type in PG(2, 27)

De Clerck et al. (J Comb Theory, 2011) counted the number of non-isomorphic Mathon maximal arcs of degree-8 in PG(2, 2 h ), h ≠ 7 and prime. In this article we will show that in PG(2, 27) a special class of Mathon maximal arcs of degree-8 arises which admits a Singer group (i.e. a sharply transitive group) on the 7 conics of these arcs. We will give a detailed description of these arcs, and then count the total number of non-isomorphic Mathon maximal arcs of degree-8. Finally we show that the special arcs found in PG(2, 27) extend to two infinite families of Mathon arcs of degree-8 in PG(2, 2 k ), k odd and divisible by 7, while maintaining the nice property of admitting a Singer group.