Finite Simple Graph Research Articles

Decomposition of complete multigraphs into crown graphs

Let λKv be the complete multigraph, G a finite simple graph. A G-design of λKv is denoted by GD(v,G,λ). The crown graph Qn is obtained by joining single pendant edge to each vertex of an n-cycle. We give new constructions for Qn-designs. Let v and λ be two positive integers. For n=4, 6, 8 and λ≥1, there exists a GD(v,Qn,λ) if and only if either (1) v>2n and λv(v−1)≡0 (mod 4n), or (2) v=2n and λ≡0 (mod 4). Let n≥4 be even. Then (1) there exists a GD(2n,Qn,λ) if and only if λ≡0 (mod 4). (2) There exists a GD(2n+1,Qn,λ) when λ≡0 (mod 4).

Kneser Representations of Graphs

The Kneser graph $K_{n:k}$ for positive integers $n\ge k$ has as its vertex set the $k$-element subsets of some $n$-set, with disjoint sets being adjacent. Every finite simple graph can be found as an induced subgraph of some Kneser graph; this article explores some questions arising from that fact.

Zeros of semilinear systems with applications to nonlinear partial difference equations on graphs†

Let Q be a m × m real matrix and f j : ℝ → ℝ, j = 1, …, m, be some given functions. If x and f(x) are column vectors whose j-coordinates are x j and f j (x j ), respectively, then we apply the finite dimensional version of the mountain pass theorem to provide conditions for the existence of solutions of the semilinear system Qx = f(x) for Q symmetric and positive semi-definite. The arguments we use are a simple adaptation of the ones used by Neuberger. An application of the above concerns partial difference equations on a finite, connected simple graph. A derivation of a graph 𝒢 is just any linear operator D:C 0(𝒢) → C 0(𝒢), where C 0(𝒢) is the real vector space of real maps defined on the vertex set V of the graph. Given a derivation D and a function F:V × ℝ → ℝ, one has associated a partial difference equation Dμ = F(v,μ), and one searches for solutions μ ∈ C 0(𝒢). Sufficient conditions in order to have non-trivial solutions of partial difference equations on any finite, connected simple graph for D symmetric and positive semi-definite derivation are provided. A metric (or weighted) graph is a pair (𝒢, d), where 𝒢 is a connected finite degree simple graph and d is a positive function on the set of edges of the graph. The metric d permits to consider some classical derivations, such as the Laplacian operator ▵2. In (Neuberger, Elliptic partial difference equations on graphs, Experiment. Math. 15 (2006), pp. 91–107) was considered the nonlinear elliptic partial difference equations ▵2 u = F(u), for the metric d = 1.

About acyclic edge colourings of planar graphs

Let G = ( V , E ) be any finite simple graph. A mapping C : E → [ k ] is called an acyclic edge k-colouring of G, if any two adjacent edges have different colours and there are no bichromatic cycles in G. In other words, for every pair of distinct colours i and j, the subgraph induced by all the edges which have either colour i or j is acyclic. The smallest number k of colours, such that G has an acyclic edge k-colouring is called the acyclic chromatic index of G and is denoted by χ a ′ ( G ) . In 1991, Alon et al. [N. Alon, C.J.H. McDiarmid, B.A. Reed, Acyclic coloring of graphs, Random Structures and Algorithms 2 (1991) 277–288] proved that χ a ′ ( G ) ⩽ 64 Δ ( G ) for any graph G of maximum degree Δ ( G ) . This bound was later improved to 16 Δ ( G ) by Molloy and Reed in [M. Molloy, B. Reed, Further algorithmic aspects of the local lemma, in: Proceedings of the 30th Annual ACM Symposium on Theory of Computing, 1998, pp. 524–529]. In this paper we prove that χ a ′ ( G ) ⩽ Δ ( G ) + 6 for a planar graph G without cycles of length three and that the same holds if G has an edge-partition into two forests. We also show that χ a ′ ( G ) ⩽ 2 Δ ( G ) + 29 if G is planar.

A simple criterion on degree sequences of graphs

A finite sequence of nonnegative integers is called graphic if the terms in the sequence can be realized as the degrees of vertices of a finite simple graph. We present two new characterizations of graphic sequences. The first of these is similar to a result of Havel-Hakimi, and the second equivalent to a result of Erdős & Gallai, thus providing a short proof of the latter result. We also show how some known results concerning degree sets and degree sequences follow from our results.

A new class of antimagic Cartesian product graphs

An antimagic labeling of a finite undirected simple graph with m edges and n vertices is a bijection from the set of edges to the integers 1 , … , m such that all n -vertex sums are pairwise distinct, where a vertex sum is the sum of labels of all edges incident with the same vertex. A graph is called antimagic if it has an antimagic labeling. In 1990, Hartsfield and Ringel [N. Hartsfield, G. Ringel, Pearls in Graph Theory, Academic Press, INC., Boston, 1990, pp. 108–109, Revised version, 1994] conjectured that every simple connected graph, except K 2 , is antimagic. In this article, we prove that a new class of Cartesian product graphs are antimagic. In particular, by combining this result and the antimagicness result on toroidal grids (Cartesian products of two cycles) in [Tao-Ming Wang, Toroidal grids are anti-magic, in: Proc. 11th Annual International Computing and Combinatorics Conference COCOON’2005, in: LNCS, vol. 3595, Springer, 2005, pp. 671–679], all Cartesian products of two or more regular graphs of positive degree can be proved to be antimagic.

A result on combinatorial curvature for embedded graphs on a surface

Let G be an infinite graph embedded in a surface such that each open face of the embedding is homeomorphic to an open disk and is bounded by finite number of edges. For each vertex x of G , we define the combinatorial curvature K G ( x ) = 1 − d ( x ) 2 + ∑ σ ∈ F ( x ) 1 | σ | , where d ( x ) is the degree of x , F ( x ) is the multiset of all open faces σ in the embedding such that the closure σ ̄ contains x , and | σ | is the number of sides of edges bounding the face σ . In this paper, for a finite simple graph G embedded in a surface with 3 ≤ d G ( x ) < ∞ and K G ( x ) > 0 for all x ∈ V ( G ) , we have (i) if G is embedded in a projective plane and | V ( G ) | = n ≥ 290 , then G is isomorphic to P n ; (ii) if G is embedded in a sphere and | V ( G ) | = n ≥ 580 , then G is isomorphic to either A n or B n .

Eriksson’s numbers game and finite Coxeter groups

The numbers game is a one-player game played on a finite simple graph with certain “amplitudes” assigned to its edges and with an initial assignment of real numbers to its nodes. The moves of the game successively transform the numbers at the nodes using the amplitudes in a certain way. This game and its interactions with Coxeter/Weyl group theory and Lie theory have been studied by many authors. In particular, Eriksson connects certain geometric representations of Coxeter groups with games on graphs with certain real number amplitudes. Games played on such graphs are “E-games”. Here we investigate various finiteness aspects of E-game play: We extend Eriksson’s work relating moves of the game to reduced decompositions of elements of a Coxeter group naturally associated to the game graph. We use Stembridge’s theory of fully commutative Coxeter group elements to classify what we call here the “adjacency-free” initial positions for finite E-games. We characterize when the positive roots for certain geometric representations of finite Coxeter groups can be obtained from E-game play. Finally, we provide a new Dynkin diagram classification result of E-game graphs meeting a certain finiteness requirement.

On anti-magic labeling for graph products

An anti-magic labeling of a finite simple undirected graph with p vertices and q edges is a bijection from the set of edges to the set of integers { 1 , 2 , … , q } such that the vertex sums are pairwise distinct, where the vertex sum at one vertex is the sum of labels of all edges incident to such vertex. A graph is called anti-magic if it admits an anti-magic labeling. Hartsfield and Ringel conjectured in 1990 that all connected graphs except K 2 are anti-magic. Recently, Alon et al. showed that this conjecture is true for dense graphs, i.e. it is true for p-vertex graphs with minimum degree Ω ( log p ) . In this article, new classes of sparse anti-magic graphs are constructed through Cartesian products and lexicographic products.

Bertrand's postulate, the prime number theorem and product anti-magic graphs

Let the edges of a finite simple graph G = ( V , E ) , | V | = n , | E | = m , be labeled by 1 , 2 , … , m . Denote by w ( u ) the product of all the labels of edges incident with a vertex u . The graph G is called product anti-magic if it is possible that the above labeling results in all values w ( u ) being distinct. Following an old conjecture of Hartsfield and Ringel on (sum) anti-magic graphs (see [N. Hartsfield and G. Ringel, Pearls in Graph Theory, Academic Press, Inc., Boston, 1990, pp. 108–109 (revised version, 1994)]), Figueroa-Centeno et al. [Bertrand's postulate and magical product labellings, Bull. ICA 30 (2000) 53–65] conjectured that every connected graph of size m is product anti-magic iff m ⩾ 3 . In this paper we prove this conjecture for dense graphs, complete multi-partite graphs and some other families of graphs.

Quasi-Kähler Bestvina-Brady groups

A finite simple graph Γ determines a right-angled Artin group GΓ, with one generator for each vertex v, and with one commutator relation vw = wv for each pair of vertices joined by an edge. The Bestvina-Brady group NΓ is the kernel of the projection GΓ → Z, which sends each generator v to 1. We establish precisely which graphs Γ give rise to quasi-Kahler (respectively, Kahler) groups NΓ. This yields examples of quasi-projective groups which are not commensurable (up to finite kernels) to the fundamental group of any aspherical, quasiprojective variety.

Lattice grids and prisms are antimagic

An antimagic labelling of a finite undirected simple graph with m edges and n vertices is a bijection from the set of edges to the integers 1 , … , m such that all n vertex sums are pairwise distinct, where a vertex sum is the sum of labels of all edges incident with the same vertex. A graph is called antimagic if it has an antimagic labelling. In 1990, Hartsfield and Ringel conjectured that every connected graph, but K 2 , is antimagic. In [T.-M. Wang, Toroidal grids are antimagic, in: Proc. 11th Annual International Computing and Combinatorics Conference, COCOON’2005, in: LNCS, vol. 3595, Springer, 2005, pp. 671–679], Wang showed that the toroidal grids (the Cartesian products of two or more cycles) are antimagic. Two open problems left in Wang’s paper are about the antimagicness of lattice grid graphs and prism graphs, which are the Cartesian products of two paths, and of a cycle and a path, respectively. In this article, we prove that these two classes of graphs are antimagic, by constructing such antimagic labellings.

A short proof of a theorem on degree sets of graphs

For a finite simple graph G, we denote the set of degrees of its vertices, known as its degree set, by D ( G ) . Kapoor, Polimeni and Wall [Degree sets for graphs, Fund. Math. 95 (1977) 189–194] have determined the least number of vertices among graphs with a given degree set. We give a very short proof of this result.

The deficiency of a regular graph

A proper edge coloring c : E ( G ) → Z of a finite simple graph G is an interval coloring if the colors used at each vertex form a consecutive interval of integers. Many graphs do not have interval colorings, and the deficiency of a graph is an invariant that measures how close a graph comes to having an interval coloring. In this paper we search for tight upper bounds on the deficiencies of k-regular graphs in terms of the number of vertices. We find exact values for 1 ⩽ k ⩽ 4 and bounds for larger k.

A constructive characterization of 3-connected triangle-free graphs

We present a method to construct the class C of finite simple 3-connected triangle-free graphs from the 3-regular complete bipartite graph K 3 , 3 and the skeleton of a 3-dimensional cube by sequentially applying one of five operations without leaving C in the intermediate steps. Each of these operations can be considered as a sequence of at most six vertex splittings and edge additions.

On the least size of a graph with a given degree set

The degree set of a finite simple graph G is the set of distinct degrees of vertices of G. A theorem of Kapoor et al. [Degree sets for graphs, Fund. Math. 95 (1977) 189–194] asserts that the least order of a graph with a given degree set D is 1 + max ( D ) . We look at the analogous problem concerning the least size of a graph with a given degree set D . We determine the least size for the sets D when (i) | D | ⩽ 3 ; (ii) D = { 1 , 2 , … , n } ; and (iii) every element in D is at least | D | . In addition, we give sharp upper and lower bounds in all cases.

Algebraic invariants for right-angled Artin groups

A finite simplicial graph Γ determines a right-angled Artin group GΓ, with generators corresponding to the vertices of Γ, and with a relation υw=wυ for each pair of adjacent vertices. We compute the lower central series quotients, the Chen quotients, and the (first) resonance variety of GΓ, directly from the graph Γ.

Generation of simple quadrangulations of the sphere

A simple quadrangulation of the sphere is a finite simple graph embedded on the sphere such that every face is bounded by a walk of 4 edges. We consider the following classes of simple quadrangulations: arbitrary, minimum degree 3, 3-connected, and 3-connected without non-facial 4-cycles. In each case, we show how the class can be generated by starting with some basic graphs in the class and applying a sequence of local modifications. The duals of our algorithms generate classes of quartic (4-regular) planar graphs. In the case of minimum degree 3, our result is a strengthening of a theorem of Nakamoto and almost implicit in Nakamoto's proof. In the case of 3-connectivity, a corollary of our theorem matches a theorem of Batagelj. However, Batagelj's proof contained a serious error which cannot easily be corrected. We also give a theoretical enumeration of rooted planar quadrangulations of minimum degree 3, and some counts obtained by a program of Brinkmann and McKay that implements our algorithm.

On Compact Graphs

Let G be a finite simple graph with adjacency matrix A, and let P(A) be the convex closure of the set of all permutation matrices commuting with A. G is said to be compact if every doubly stochastic matrix which commutes with A is in P(A). In this paper, we characterize 3–regular compact graphs and prove that if G is a connected regular compact graph, G – v is also compact, and give a family of almost regular compact connected graphs.

Algebraic Characterizations of Graph Regularity Conditions

It is well-known that a connected finite simple graph is regular if and only if the all-ones matrix spans an ideal of its adjacency algebra. We show that several other graph regularity conditions involving pairs and triples of vertices also have ideal theoretic characterizations in some appropriate algebras.