ON A CONJECTURE ON nTH ORDER DEGREE REGULAR GRAPHS

Abstract

For n a positive integer and v a vertex of a graph G, the nth order degree of v in G, denoted by deg_nv, is the number of vertices at distance n from v. The graph G is said to be nth order regular of degree k if, for every vertex v of G, deg_nv = k. The following conjecture due to Alavi, Lick, and Zou is proved: For n ≥ 2, if G is a connected nth order regular graph of degree 1, then G is either a path of length 2n—1 or G has diameter n. Properties of nth order regular graphs of degree k, k ≥ 1, are investigated.     

Graham’s pebbling conjecture on product of thorn graphs of complete graphs

The pebbling number of a graph G, f(G), is the least n such that, no matter how n pebbles are placed on the vertices of G, we can move a pebble to any vertex by a sequence of pebbling moves, each move taking two pebbles off one vertex and placing one on an adjacent vertex. Let p₁,p₂,…,p_n be positive integers and G be such a graph, V(G)=n. The thorn graph of the graph G, with parameters p₁,p₂,…,p_n, is obtained by attaching p_i new vertices of degree 1 to the vertex u_i of the graph G, i=1,2,…,n. Graham conjectured that for any connected graphs G and H, f(G×H)≤f(G)f(H). We show that Graham’s conjecture holds true for a thorn graph of the complete graph with every by a graph with the two-pebbling property. As a corollary, Graham’s conjecture holds when G and H are the thorn graphs of the complete graphs with every .     

Gorenstein binomial edge ideals associated with scrolls

Let I_G be the binomial edge ideal on the generic 2×n - Hankel matrix associated with a closed graph G on the vertex set [n]. We characterize the graphs G for which I_G has maximal regularity and is Gorenstein.     

On the rank spread of graphs

For a simple graph G?=?(𝒱, ?) with vertex-set 𝒱?=?{1,?…?,?n}, let 𝒮(G) be the set of all real symmetric n-by-n matrices whose graph is G. We present terminology linking established as well as new results related to the minimum rank problem, with spectral properties in graph theory. The minimum rank mr(G) of G is the smallest possible rank over all matrices in 𝒮(G). The rank spread r _v (G) of G at a vertex v, defined as mr(G)???mr(G???v), can take values ??∈?{0,?1,?2}. In general, distinct vertices in a graph may assume any of the three values. For ??=?0 or 1, there exist graphs with uniform r _v (G) (equal to the same integer at each vertex v). We show that only for ??=?0, will a single matrix A in 𝒮(G) determine when a graph has uniform rank spread. Moreover, a graph G, with vertices of rank spread zero or one only, is a λ-core graph for a λ-optimal matrix A in 𝒮(G). We also develop sufficient conditions for a vertex of rank spread zero or two and a necessary condition for a vertex of rank spread two.     

Connected graphs with maximal Q-index: The one-dominating-vertex case

By the signless Laplacian of a (simple) graph G we mean the matrix Q(G)=D(G)+A(G), where A(G),D(G) denote respectively the adjacency matrix and the diagonal matrix of vertex degrees of G. For every pair of positive integers n,k, it is proved that if 3?k?n-3, then H_n,k, the graph obtained from the star K_1,n-1 by joining a vertex of degree 1 to k+1 other vertices of degree 1, is the unique connected graph that maximizes the largest signless Laplacian eigenvalue over all connected graphs with n vertices and n+k edges.     

A Classification of the Strongly Regular Generalised Johnson Graphs

The generalised Johnson graphs are the graphs J(n, k, m) whose vertices are the k subsets of {1, 2, . . . , n}, with two vertices J ₁ and J ₂ joined by an edge if and only if ${{|J_1 \cap J_2| = m}}$ . A graph is called d-regular if every vertex has exactly d edges incident to it. A d-regular graph on v vertices is called a (v, d, a, c)-strongly regular graph if every pair of adjacent vertices have exactly a common neighbours and every pair of non-adjacent vertices have exactly c common neighbours. The triangular graphs J(n, 2, 1), their complements J(n, 2, 0), the sporadic examples J(10, 3, 1) and J(7, 3, 1), as well as the trivially strongly regular graphs J(2k, k, 0) are examples of strongly regular generalised Johnson graphs. In this paper we prove that there are no other strongly regular generalised Johnson graphs.     

Spectra of Some Interesting Combinatorial Matrices Related to Oriented Spanning Trees on a Directed Graph

The Laplacian of a directed graph G is the matrix L(G) = O(G) –, A(G) where A(G) is the adjaceney matrix of G and O(G) the diagonal matrix of vertex outdegrees. The eigenvalues of G are the eigenvalues of A(G). Given a directed graph G we construct a derived directed graph D(G) whose vertices are the oriented spanning trees of G. Using a counting argument, we describe the eigenvalues of D(G) and their multiplicities in terms of the eigenvalues of the induced subgraphs and the Laplacian matrix of G. Finally we compute the eigenvalues of D(G) for some specific directed graphs G. A recent conjecture of Propp for D(H _n ) follows, where H _n stands for the complete directed graph on n vertices without loops.     

Random Graph Coverings I: General Theory and Graph Connectivity

In this paper we describe a simple model for random graphs that have an n-fold covering map onto a fixed finite base graph. Roughly, given a base graph G and an integer n, we form a random graph by replacing each vertex of G by a set of n vertices, and joining these sets by random matchings whenever the corresponding vertices are adjacent in G. The resulting graph covers the original graph in the sense that the two are locally isomorphic. We suggest possible applications of the model, such as constructing graphs with extremal properties in a more controlled fashion than offered by the standard random models, and also "randomizing" given graphs. The main specific result that we prove here (Theorem 1) is that if is the smallest vertex degree in G, then almost all n-covers of G are -connected. In subsequent papers we will address other graph properties, such as girth, expansion and chromatic number. Received June 21, 1999/Revised November 16, 2000 RID="*" ID="*" Work supported in part by grants from the Israel Academy of Aciences and the Binational Israel-US Science Foundation.     

Proof of a tiling conjecture of Komlós

A conjecture of Komlós states that for every graph H, there is a constant K such that if G is any n‐vertex graph of minimum degree at least (1 ? (1/χ_cr(H)))n, where χ_cr(H) denotes the critical chromatic number of H, then G contains an H‐matching that covers all but at most K vertices of G. In this paper we prove that the conjecture holds for all sufficiently large values of n. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 23: 180–205, 2003     

On antimagic directed graphs

An antimagic labeling of an undirected graph G with n vertices and m edges is a bijection from the set of edges of G 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 that vertex. A graph is called antimagic if it admits an antimagic labeling. In (N. Hartsfield and G. Ringel, Pearls in Graph Theory, Academic Press, Boston, 1990, pp. 108–109), Hartsfield and Ringel conjectured that every simple connected graph, other than K₂, is antimagic. Despite considerable effort in recent years, this conjecture is still open. In this article we study a natural variation; namely, we consider antimagic labelings of directed graphs. In particular, we prove that every directed graph whose underlying undirected graph is “dense” is antimagic, and that almost every undirected d‐regular graph admits an orientation which is antimagic. © 2009 Wiley Periodicals, Inc. J Graph Theory 64: 219–232, 2010     

Hamiltonian Cycles in Almost Claw-Free Graphs

 Some known results on claw-free graphs are generalized to the larger class of almost claw-free graphs. In this paper, we prove the following two results and conjecture that every 5-connected almost claw-free graph is hamiltonian. (1). Every 2-connected almost claw-free graph G∉J on n≤ 4 δ vertices is hamiltonian, where J is the set of all graphs defined as follows: any graph G in J can be decomposed into three disjoint connected subgraphs G ₁, G ₂ and G ₃ such that E _G (G _i , G _j ) = {u _i , u _j , v _i v _j } for i≠j and i,j = 1, 2, 3 (where u _i ≠v _i ∈V(G _i ) for i = 1, 2, 3). Moreover the bound 4δ is best possible, thereby fully generalizing several previous results. (2). Every 3-connected almost claw-free graph on at most 5δ−5 vertices is hamiltonian, hereby fully generalizing the corresponding result on claw-free graphs. Received: September 21, 1998 Final version received: August 18, 1999     

Every 3‐connected claw‐free Z8‐free graph is Hamiltonian

A biclique of a graph G is a maximal induced complete bipartite subgraph of G. Given a graph G, the biclique matrix of G is a {0,1,?1} matrix having one row for each biclique and one column for each vertex of G, and such that a pair of 1, ?1 entries in a same row corresponds exactly to adjacent vertices in the corresponding biclique. We describe a characterization of biclique matrices, in similar terms as those employed in Gilmore's characterization of clique matrices. On the other hand, the biclique graph of a graph is the intersection graph of the bicliques of G. Using the concept of biclique matrices, we describe a Krausz‐type characterization of biclique graphs. Finally, we show that every induced P₃ of a biclique graph must be included in a diamond or in a 3‐fan and we also characterize biclique graphs of bipartite graphs. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 1–16, 2010     

Another two graphs with no planar covers

A graph H is a cover of a graph G if there exists a mapping φ from V( H ) onto V( G ) such that φ maps the neighbors of every vertex υ in H bijectively to the neighbors of φ(υ) in G . Negami conjectured in 1986 that a connected graph has a finite planar cover if and only if it embeds in the projective plane. It follows from the results of Archdeacon, Fellows, Negami, and the author that the conjecture holds as long as K _1,2,2,2 has no finite planar cover. However, this is still an open question, and K _1,2,2,2 is not the only minor‐minimal graph in doubt. Let ??₄ (?₂) denote the graph obtained from K _1,2,2,2 by replacing two vertex‐disjoint triangles (four edge‐disjoint triangles) not incident with the vertex of degree 6 with cubic vertices. We prove that the graphs ??₄ and ?₂ have no planar covers. This fact is used in [P. Hlin?ný, R. Thomas, On possible counterexamples to Negami's planar cover conjecture, 1999 (submitted)] to show that there are, up to obvious constructions, at most 16 possible counterexamples to Negami's conjecture. © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 227–242, 2001     

Antimagic labelling of vertex weighted graphs

Suppose G is a graph, k is a non‐negative integer. We say G is k‐antimagic if there is an injection f: E→{1, 2, …, |E| + k} such that for any two distinct vertices u and v, . We say G is weighted‐k‐antimagic if for any vertex weight function w: V→?, there is an injection f: E→{1, 2, …, |E| + k} such that for any two distinct vertices u and v, . A well‐known conjecture asserts that every connected graph G≠K₂ is 0‐antimagic. On the other hand, there are connected graphs G≠K₂ which are not weighted‐1‐antimagic. It is unknown whether every connected graph G≠K₂ is weighted‐2‐antimagic. In this paper, we prove that if G has a universal vertex, then G is weighted‐2‐antimagic. If G has a prime number of vertices and has a Hamiltonian path, then G is weighted‐1‐antimagic. We also prove that every connected graph G≠K₂ on n vertices is weighted‐ ?3n/2?‐antimagic. Copyright © 2011 Wiley Periodicals, Inc. J Graph Theory     

Matroidal families of finite connected nonhomeomorphic graphs exist

A matroidal family is a nonempty set ? of connected finite graphs such that for every arbitrary finite graph G the edge sets of the subgraphs of G which are isomorphic to an element of ? form a matroid on the edge set of G. In the present paper the question whether there are any matroidal families besides the four previously described by Simões-Pereira is answered affirmatively. It is shown that for every natural number n ? 2 there is a matroidal family that contains the complete graph with n vertices. For n = 4 this settles Simões-Pereira's conjecture that there is a matroidal family containing all wheels.     

An edge-deletion problem for locally finite graphs

For each positive integer n, let T_n be the tree in which exactly one vertex has degree n and all the other vertices have degree n + 1. A graph G is called stable if its edge set is nonempty and if deleting an arbitrary edge of G there is always a component of the residue graph which is isomorphic to G. The question whether there are locally finite stable graphs that are not isomorphic to one of the graphs T_n is answered affirmatively by constructing an uncountable family of pairwise nonisomorphic, locally finite, stable graphs. Further, the following results are proved: (1) Among the locally finite trees containing no subdivision of T₂, the oneway infinite path T₁ is the only stable graph. (2) Among the locally finite graphs containing no two-way infinite path, T₁ is also the only stable graph.     

On constructible graphs,locally Helly graphs,and convexity

A (finite or infinite) graph G is constructible if there exists a well‐ordering ≤ of its vertices such that for every vertex x which is not the smallest element, there is a vertex y < x which is adjacent to x and to every neighbor z of x with z < x. Particular constructible graphs are Helly graphs and connected bridged graphs. In this paper we study a new class of constructible graphs, the class of locally Helly graphs. A graph G is locally Helly if, for every pair (x,y) of vertices of G whose distance is d ≥ 2, there exists a vertex whose distance to x is d ? 1 and which is adjacent to y and to all neighbors of y whose distance to x is at most d. Helly graphs are locally Helly, and the converse holds for finite graphs. Among different properties we prove that a locally Helly graph is strongly dismantable, hence cop‐win, if and only if it contains no isometric rays. We show that a locally Helly graph G is finitely Helly, that is, every finite family of pairwise non‐disjoint balls of G has a non‐empty intersection. We give a sufficient condition by forbidden subgraphs so that the three concepts of Helly graphs, of locally Helly graphs and of finitely Helly graphs are equivalent. Finally, generalizing different results, in particular those of Bandelt and Chepoi 1 about Helly graphs and bridged graphs, we prove that the Helly number h(G) of the geodesic convexity in a constructible graph G is equal to its clique number ω(G), provided that ω(G) is finite. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 280–298, 2003     

Quasi‐randomness of graph balanced cut properties

Quasi‐random graphs can be informally described as graphs whose edge distribution closely resembles that of a truly random graph of the same edge density. Recently, Shapira and Yuster proved the following result on quasi‐randomness of graphs. Let k ≥ 2 be a fixed integer, α₁,…,α_k be positive reals satisfying \begin{align*}\sum_{i} \alpha_i = 1\end{align*} and (α₁,…,α_k)≠(1/k,…,1/k), and G be a graph on n vertices. If for every partition of the vertices of G into sets V ₁,…,V _k of size α₁n,…,α_kn, the number of complete graphs on k vertices which have exactly one vertex in each of these sets is similar to what we would expect in a random graph, then the graph is quasi‐random. However, the method of quasi‐random hypergraphs they used did not provide enough information to resolve the case (1/k,…,1/k) for graphs. In their work, Shapira and Yuster asked whether this case also forces the graph to be quasi‐random. Janson also posed the same question in his study of quasi‐randomness under the framework of graph limits. In this paper, we positively answer their question. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011     

On minimally b-imperfect graphs

A b-coloring is a coloring of the vertices of a graph such that each color class contains a vertex that has a neighbour in all other color classes. The b-chromatic number of a graph G is the largest integer k such that G admits a b-coloring with k colors. A graph is b-perfect if the b-chromatic number is equal to the chromatic number for every induced subgraph H of G. A graph is minimally b-imperfect if it is not b-perfect and every proper induced subgraph is b-perfect. We give a list of minimally b-imperfect graphs, conjecture that a graph is b-perfect if and only if it does not contain a graph from this list as an induced subgraph, and prove this conjecture for diamond-free graphs, and graphs with chromatic number at most three.     

Small cycle double covers of 4-connected planar graphs

Acycle double cover of a graph,G, is a collection of cycles,C, such that every edge ofG lies in precisely two cycles ofC. TheSmall Cycle Double Cover Conjecture, proposed by J. A. Bondy, asserts that every simple bridgeless graph onn vertices has a cycle double cover with at mostn–1 cycles, and is a strengthening of the well-knownCycle Double Cover Conjecture. In this paper, we prove Bondy's conjecture for 4-connected planar graphs.