Sphericity, cubicity, and edge clique covers of graphs

AbstractThe sphericity sph(G) of a graph G is the minimum dimension d for which G is the intersection graph of a family of congruent spheres in Rd. The edge clique cover number θ(G) is the minimum cardinality of a set of cliques (complete subgraphs) that covers all edges of G. We prove that if G has at least one edge, then sph(G)⩽θ(G). Our upper bound remains valid for intersection graphs defined by balls in the Lp-norm for 1⩽p⩽∞

Cubicity of interval graphs and the claw number

Let $G(V,E)$ be a simple, undirected graph where $V$ is the set of vertices and $E$ is the set of edges. A $b$-dimensional cube is a Cartesian product $I_1\times I_2\times...\times I_b$, where each $I_i$ is a closed interval of unit length on the real line. The \emph{cubicity} of $G$, denoted by \cub(G) is the minimum positive integer $b$ such that the vertices in $G$ can be mapped to axis parallel $b$-dimensional cubes in such a way that two vertices are adjacent in $G$ if and only if their assigned cubes intersect. Suppose $S(m)$ denotes a star graph on $m+1$ nodes. We define \emph{claw number} $\psi(G)$ of the graph to be the largest positive integer $m$ such that $S(m)$ is an induced subgraph of $G$. It can be easily shown that the cubicity of any graph is at least \ceil{\log_2\psi(G)}. In this paper, we show that, for an interval graph $G$ \ceil{\log_2\psi(G)}\le\cub(G)\le\ceil{\log_2\psi(G)}+2. Till now we are unable to find any interval graph with \cub(G)>\ceil{\log_2\psi(G)}. We also show that, for an interval graph $G$, \cub(G)\le\ceil{\log_2\alpha}, where $\alpha$ is the independence number of $G$. Therefore, in the special case of $\psi(G)=\alpha$, \cub(G) is exactly \ceil{\log_2\alpha}. The concept of cubicity can be generalized by considering boxes instead of cubes. A $b$-dimensional box is a Cartesian product $I_1\times I_2\times...\times I_b$, where each $I_i$ is a closed interval on the real line. The \emph{boxicity} of a graph, denoted $box(G)$, is the minimum $k$ such that $G$ is the intersection graph of $k$-dimensional boxes. It is clear that box(G)\le\cub(G). From the above result, it follows that for any graph $G$, \cub(G)\le box(G)\ceil{\log_2\alpha}

Cubicity, Degeneracy, and Crossing Number

A $k$-box $B=(R_1,...,R_k)$, where each $R_i$ is a closed interval on the real line, is defined to be the Cartesian product $R_1\times R_2\times ...\times R_k$. If each $R_i$ is a unit length interval, we call $B$ a $k$-cube. Boxicity of a graph $G$, denoted as \boxi(G), is the minimum integer $k$ such that $G$ is an intersection graph of $k$-boxes. Similarly, the cubicity of $G$, denoted as \cubi(G), is the minimum integer $k$ such that $G$ is an intersection graph of $k$-cubes. It was shown in [L. Sunil Chandran, Mathew C. Francis, and Naveen Sivadasan: Representing graphs as the intersection of axis-parallel cubes. MCDES-2008, IISc Centenary Conference, available at CoRR, abs/cs/ 0607092, 2006.] that, for a graph $G$ with maximum degree $\Delta$, \cubi(G)\leq \lceil 4(\Delta +1)\log n\rceil. In this paper, we show that, for a $k$-degenerate graph $G$, \cubi(G) \leq (k+2) \lceil 2e \log n \rceil. Since $k$ is at most $\Delta$ and can be much lower, this clearly is a stronger result. This bound is tight. We also give an efficient deterministic algorithm that runs in $O(n^2k)$ time to output a $8k(\lceil 2.42 \log n\rceil + 1)$ dimensional cube representation for $G$. An important consequence of the above result is that if the crossing number of a graph $G$ is $t$, then \boxi(G) is $O(t^{1/4}{\lceil\log t\rceil}^{3/4})$ . This bound is tight up to a factor of $O((\log t)^{1/4})$. We also show that, if $G$ has $n$ vertices, then \cubi(G) is $O(\log n + t^{1/4}\log t)$. Using our bound for the cubicity of $k$-degenerate graphs we show that cubicity of almost all graphs in $\mathcal{G}(n,m)$ model is $O(d_{av}\log n)$, where $d_{av}$ denotes the average degree of the graph under consideration.Comment: 21 page

OV Graphs Are (Probably) Hard Instances

© Josh Alman and Virginia Vassilevska Williams. A graph G on n nodes is an Orthogonal Vectors (OV) graph of dimension d if there are vectors v1, . . ., vn ∈ {0, 1}d such that nodes i and j are adjacent in G if and only if hvi, vji = 0 over Z. In this paper, we study a number of basic graph algorithm problems, except where one is given as input the vectors defining an OV graph instead of a general graph. We show that for each of the following problems, an algorithm solving it faster on such OV graphs G of dimension only d = O(log n) than in the general case would refute a plausible conjecture about the time required to solve sparse MAX-k-SAT instances: Determining whether G contains a triangle. More generally, determining whether G contains a directed k-cycle for any k ≥ 3. Computing the square of the adjacency matrix of G over Z or F2. Maintaining the shortest distance between two fixed nodes of G, or whether G has a perfect matching, when G is a dynamically updating OV graph. We also prove some complementary results about OV graphs. We show that any problem which is NP-hard on constant-degree graphs is also NP-hard on OV graphs of dimension O(log n), and we give two problems which can be solved faster on OV graphs than in general: Maximum Clique, and Online Matrix-Vector Multiplication

Edge Clique Cover of Claw-free Graphs

The smallest number of cliques, covering all edges of a graph $G$, is called the (edge) clique cover number of $G$ and is denoted by $cc(G)$. It is an easy observation that for every line graph $G$ with $n$ vertices, $cc(G)\leq n$. G. Chen et al. [Discrete Math. 219 (2000), no. 1--3, 17--26; MR1761707] extended this observation to all quasi-line graphs and questioned if the same assertion holds for all claw-free graphs. In this paper, using the celebrated structure theorem of claw-free graphs due to Chudnovsky and Seymour, we give an affirmative answer to this question for all claw-free graphs with independence number at least three. In particular, we prove that if $G$ is a connected claw-free graph on $n$ vertices with $\alpha(G)\geq 3$, then $cc(G)\leq n$ and equality holds if and only if $G$ is either the graph of icosahedron, or the complement of a graph on $10$ vertices called twister or the $p^{th}$ power of the cycle $C_n$, for $1\leq p \leq \lfloor (n-1)/3\rfloor$.Comment: 74 pages, 4 figure