Note on the Pair-crossing Number and the Odd-crossing Number

The crossing number of a graph G is the minimum possible number of edge-crossings in a drawing of G, the pair-crossing number is the minimum possible number of crossing pairs of edges in a drawing of G, and the odd-crossing number is the minimum number of pairs of edges that cross an odd number of times. Clearly, . We construct graphs with . This improves the bound of Pelsmajer, Schaefer and Štefankovič. Our construction also answers an old question of Tutte. Slightly improving the bound of Valtr, we also show that if the pair-crossing number of G is k, then its crossing number is at most O(k ²/log ² k). G. Tóth’s research was supported by the Hungarian Research Fund grant OTKA-K-60427 and the Research Foundation of the City University of New York.     

Upper Bounds on the Paired Domination Subdivision Number of a Graph

A paired dominating set of a graph G with no isolated vertex is a dominating set S of vertices such that the subgraph induced by S in G has a perfect matching. The paired domination number of G, denoted by γ _pr(G), is the minimum cardinality of a paired dominating set of G. The paired domination subdivision number ${{\rm sd}_{\gamma _{\rm pr}}(G)}$ is the minimum number of edges to be subdivided (each edge of G can be subdivided at most once) in order to increase the paired domination number. In this paper, we show that if G is a connected graph of order at least 4, then ${{\rm sd}_{\gamma _{\rm pr}}(G)\leq 2|V(G)|-5}$ . We also characterize trees T such that ${{\rm sd}_{\gamma _{\rm pr}}(T) \geq |V(T)| /2}$ .     

Visibility Graphs of Point Sets in the Plane

The visibility graph of a discrete point set X⊂ℝ² has vertex set X and an edge xy for every two points x,y∈X whenever there is no other point in X on the line segment between x and y. We show that for every graph G, there is a point set X∈ℝ², such that the subgraph of induced by X is isomorphic to G. As a consequence, we show that there are visibility graphs of arbitrary high chromatic number with clique number 6 settling a question by Kára, Pór and Wood. Supported by the DFG Research Center Matheon (FZT86).     

Graph von Neumann Algebras

In this paper, we will define a graph von Neumann algebra over a fixed von Neumann algebra M, where G is a countable directed graph, by a crossed product algebra = M × _α , where is the graph groupoid of G and α is the graph-representation. After defining a certain conditional expectation from onto its M-diagonal subalgebra we can see that this crossed product algebra is *-isomorphic to an amalgamated free product where = vN(M × _α where is the subset of consisting of all reduced words in {e, e ^–1} and M × _α is a W ^*-subalgebra of as a new graph von Neumann algebra induced by a graph G _e . Also, we will show that, as a Banach space, a graph von Neumann algebra is isomorphic to a Banach space ⊕ where is a certain subset of the set E(G)* of all words in the edge set E(G) of G. The author really appreciates to Prof F. Radulescu and Prof P. Jorgensen for the valuable discussion and kind advice. Also, he appreciates all supports from St. Ambrose Univ.. In particular, he thanks to Prof T. Anderson and Prof V. Vega for the useful conversations and suggestions.     

On the Cubicity of Interval Graphs

A k-cube (or “a unit cube in k dimensions”) is defined as the Cartesian product where R _i (for 1 ≤ i ≤ k) is an interval of the form [a _i , a _i  + 1] on the real line. The k-cube representation of a graph G is a mapping of the vertices of G to k-cubes such that the k-cubes corresponding to two vertices in G have a non-empty intersection if and only if the vertices are adjacent. The cubicity of a graph G, denoted as cub(G), is defined as the minimum dimension k such that G has a k-cube representation. An interval graph is a graph that can be represented as the intersection of intervals on the real line - i.e., the vertices of an interval graph can be mapped to intervals on the real line such that two vertices are adjacent if and only if their corresponding intervals overlap. We show that for any interval graph G with maximum degree Δ, . This upper bound is shown to be tight up to an additive constant of 4 by demonstrating interval graphs for which cubicity is equal to .     

Disproof of a Conjecture on the Subdivision Domination Number of a Graph

A set S of vertices of a graph G = (V,E) is a dominating set if every vertex of is adjacent to some vertex in S. The domination number γ(G) is the minimum cardinality of a dominating set of G. The domination subdivision number sd_γ(G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the domination number. Haynes et al. (Discussiones Mathematicae Graph Theory 21 (2001) 239-253) conjectured that for any graph G with . In this note we first give a counterexample to this conjecture in general and then we prove it for a particular class of graphs.     

Fredholm Properties of Band-dominated Operators on Periodic Discrete Structures

Let (X, ~) be a combinatorial graph the vertex set X of which is a discrete metric space. We suppose that a discrete group G acts freely on (X, ~) and that the fundamental domain with respect to the action of G contains only a finite set of points. A graph with these properties is called periodic with respect to the group G. We examine the Fredholm property and the essential spectrum of band-dominated operators acting on the spaces l ^p (X) or c_0(X), where (X, ~) is a periodic graph. Our approach is based on the thorough use of band-dominated operators. It generalizes the necessary and sufficient results obtained in [39] in the special case and in [42] in case X = G is a general finitely generated discrete group. Submitted: May 21, 2007. Revised: September 25, 2007. Accepted: November 5, 2007.     

Complexes of Graphs with Bounded Matching Size

For positive integers k,n, we investigate the simplicial complex of all graphs G on vertex set [n] such that every matching in G has size less than k. This complex (along with other associated cell complexes) is found to be homotopy equivalent to a wedge of spheres. The number and dimension of the spheres in the wedge are determined, and (partially conjectural) links to other combinatorially defined complexes are described. In addition we study for positive integers r,s and k the simplicial complex of all bipartite graphs G on bipartition such that there is no matching of size k in G, and obtain results similar to those obtained for . S. Linusson and V. Welker supported by EC’s IHRP program through grant HPRN-CT-2001-00272. J. Shareshian partially supported by National Science Foundation grants DMS-0070757 and DMS-0030483.     

The subexponentiality of products revisited

Following the work of Cline and Samorodnitsky (Stoch. Process. Their Appl. 49(1):75–98, 1994), we reexamine the subexponentiality of the product of two random variables, X and Y, which are independent and have distributions F and G, respectively. The main result is the following: If F belongs to the class [that is to say, F is subexponential and holds for some v>1] and G, with G(0–)=0 and G(0)<0, satisfies for each u>0, then the distribution of XY also belongs to the class .      

Approximation Algorithms for Maximization Problems Arising in Graph Partitioning

Given a graph G = (V, E), a weight function w: E → R⁺, and a parameter k, we consider the problem of finding a subset U  V of size k that maximizes: Max-Vertex Cover_k: the weight of edges incident with vertices in U,Max-Dense Subgraph_k: the weight of edges in the subgraph induced by U,Max-Cut_k: the weight of edges cut by the partition (U, V\U),Max-Uncut_k: the weight of edges not cut by the partition (U, V\U).For each of the above problems we present approximation algorithms based on semidefinite programming and obtain approximation ratios better than those previously published. In particular we show that if a graph has a vertex cover of size k, then one can select in polynomial time a set of k vertices that covers over 80% of the edges.     

Vertex-Colouring Edge-Weightings

A weighting w of the edges of a graph G induces a colouring of the vertices of G where the colour of vertex v, denoted c_v, is . We show that the edges of every graph that does not contain a component isomorphic to K₂ can be weighted from the set {1, . . . ,30} such that in the resulting vertex-colouring of G, for every edge (u,v) of G, c_u ≠c_v.     

Fractional f-Edge Cover Chromatic Index of Graphs

Let G be a graph, and let f be an integer function on V with ${1\leq f(v)\leq d(v)}$ to each vertex ${\upsilon \in V}$ . An f-edge cover coloring is a coloring of edges of E(G) such that each color appears at each vertex ${\upsilon \in V(G)}$ at least f(υ) times. The maximum number of colors needed to f-edge cover color G is called the f-edge cover chromatic index of G and denoted by ${\chi^{'}_{fc}(G)}$ . It is well known that any simple graph G has the f-edge cover chromatic index equal to δ _f (G) or δ _f (G) ? 1, where ${\delta_{f}(G)=\,min\{\lfloor \frac{d(v)}{f(v)} \rfloor: v\in V(G)\}}$ . The fractional f-edge cover chromatic index of a graph G, denoted by ${\chi^{'}_{fcf}(G)}$ , is the fractional f-matching number of the edge f-edge cover hypergraph ${\mathcal{H}}$ of G whose vertices are the edges of G and whose hyperedges are the f-edge covers of G. In this paper, we give an exact formula of ${\chi^{'}_{fcf}(G)}$ for any graph G, that is, ${\chi^{'}_{fcf}(G)=\,min \{\min\limits_{v\in V(G)}d_{f}(v), \lambda_{f}(G)\}}$ , where ${\lambda_{f}(G)=\min\limits_{S} \frac{|C[S]|}{\lceil (\sum\limits_{v\in S}{f(v)})/2 \rceil}}$ and the minimum is taken over all nonempty subsets S of V(G) and C[S] is the set of edges that have at least one end in S.     

On the Generalized Enveloping Algebra of a Color Lie Algebra

Let G be an abelian group, ε an anti-bicharacter of G and L a G-graded ε Lie algebra (color Lie algebra) over a field of characteristic zero. We prove that for all G-graded, positively filtered A such that the associated graded algebra is isomorphic to the G-graded ε-symmetric algebra S(L), there is a G- graded ε-Lie algebra L and a G-graded scalar two cocycle , such that A is isomorphic to U _ω (L) the generalized enveloping algebra of L associated with ω. We also prove there is an isomorphism of graded spaces between the Hochschild cohomology of the generalized universal enveloping algebra U(L) and the generalized cohomology of the color Lie algebra L. Supported by the EC project Liegrits MCRTN 505078.     

All-Pairs Small-Stretch Paths

Let G = (V, E) be a weighted undirected graph. A path between u, v  V is said to be of stretch t if its length is at most t times the distance between u and v in the graph. We consider the problem of finding small-stretch paths between all pairs of vertices in the graph G.It is easy to see that finding paths of stretch less than 2 between all pairs of vertices in an undirected graph with n vertices is at least as hard as the Boolean multiplication of two n × n matrices. We describe three algorithms for finding small-stretch paths between all pairs of vertices in a weighted graph with n vertices and m edges. The first algorithm, STRETCH₂, runs in Õ(n^3/2m^1/2) time and finds stretch 2 paths. The second algorithm, STRETCH_7/3, runs in Õ(n^7/3) time and finds stretch 7/3 paths. Finally, the third algorithm, STRETCH₃, runs in Õ(n²) and finds stretch 3 paths.Our algorithms are simpler, more efficient and more accurate than the previously best algorithms for finding small-stretch paths. Unlike all previous algorithms, our algorithms are not based on the construction of sparse spanners or sparse neighborhood covers.     

Sparse pseudo‐random graphs are Hamiltonian

In this article we study Hamilton cycles in sparse pseudo‐random graphs. We prove that if the second largest absolute value λ of an eigenvalue of a d‐regular graph G on n vertices satisfies and n is large enough, then G is Hamiltonian. We also show how our main result can be used to prove that for every c >0 and large enough n a Cayley graph X (G,S), formed by choosing a set S of c log⁵ n random generators in a group G of order n, is almost surely Hamiltonian. © 2002 Wiley Periodicals, Inc. J Graph Theory 42: 17–33, 2003     

On topological diversity of rectilinear representations of countably infinite graphs

We prove that, for each simple graph G whose set of vertices is countably infinite, there is a family ${\varvec{\mathcal{R}}(\varvec{G})}$ of the cardinality of the continuum of graphs such that (1) each graph ${\varvec{H} \in \varvec{\mathcal{R}}(\varvec{G})}$ is isomorphic to G, all vertices of H are points of the Euclidean space E ³, all edges of H are straight line segments (the ends of each edge are the vertices joined by it), the intersection of any two edges of H is either their common vertex or empty, and any isolated vertex of H does not belong to any edge of H; (2) all sets ${\varvec{\mathcal{B}}(\varvec{H})}$ ( ${\varvec{H} \in \varvec{\mathcal{R}}(\varvec{G})}$ ), where ${\varvec{\mathcal{B}}(\varvec{H})\subset \mathbf{E}^3}$ is the union of all vertices and all edges of H, are pairwise not homeomorphic; moreover, for any graphs ${\varvec{H}_1 \in \varvec{\mathcal{R}}(\varvec{G})}$ and ${\varvec{H}_2 \in \varvec{\mathcal{R}}(\varvec{G})}$ , ${\varvec{H}_1 \ne \varvec{H}_2}$ , and for any finite subsets ${\varvec{S}_i \subset \varvec{\mathcal{B}}(\varvec{H}_i)}$ (i = 1, 2), the sets ${\varvec{\mathcal{B}}(\varvec{H}_1){\setminus} \varvec{S}_1}$ and ${\varvec{\mathcal{B}}(\varvec{H}_2){\setminus} \varvec{S}_2}$ are not homeomorphic.     

Multigraphs with Δ ≥ 3 are Totally-(2Δ−1)-Choosable

The total graph T(G) of a multigraph G has as its vertices the set of edges and vertices of G and has an edge between two vertices if their corresponding elements are either adjacent or incident in G. We show that if G has maximum degree Δ(G), then T(G) is (2Δ(G) − 1)-choosable. We give a linear-time algorithm that produces such a coloring. The best previous general upper bound for Δ(G) > 3 was , by Borodin et al. When Δ(G) = 4, our algorithm gives a better upper bound. When Δ(G)∈{3,5,6}, our algorithm matches the best known bound. However, because our algorithm is significantly simpler, it runs in linear time (unlike the algorithm of Borodin et al.).     

Cutting a graph into two dissimilar halves

Given a graph G and a subset S of the vertex set of G, the discrepancy of S is defined as the difference between the actual and expected numbers of the edges in the subgraph induced on S. We show that for every graph with n vertices and e edges, n < e < n(n ? 1)/4, there is an n/2-element subset with the discrepancy of the order of magnitude of \documentclass{article}\pagestyle{empty}\begin{document}$\sqrt {ne}$\end{document} For graphs with fewer than n edges, we calculate the asymptotics for the maximum guaranteed discrepancy of an n/2-element subset. We also introduce a new notion called “bipartite discrepancy” and discuss related results and open problems.     

Strictly Deza line graphs

For a given graph G, its line graph L(G) is defined as the graph with vertex set equal to the edge set of G in which two vertices are adjacent if and only if the corresponding edges of G have exactly one common vertex. A k-regular graph of diameter 2 on υ vertices is called a strictly Deza graph with parameters (υ, k, b, a) if it is not strongly regular and any two vertices have a or b common neighbors. We give a classification of strictly Deza line graphs.     

The critical groups of a family of graphs and elliptic curves over finite fields

Let q be a power of a prime, and E be an elliptic curve defined over  . Such curves have a classical group structure, and one can form an infinite tower of groups by considering E over field extensions for all k≥1. The critical group of a graph may be defined as the cokernel of L(G), the Laplacian matrix of G. In this paper, we compare elliptic curve groups with the critical groups of a certain family of graphs. This collection of critical groups also decomposes into towers of subgroups, and we highlight additional comparisons by using the Frobenius map of E over  . This work was partially supported by the NSF, grant DMS-0500557 during the author’s graduate school at the University of California, San Diego, and partially supported by an NSF Postdoctoral Fellowship.