The multichromatic numbers of some Kneser graphs

AbstractThe Kneser graph K(m,n) has the n-subsets of {1,2,…,m} as its vertices, two such vertices being adjacent whenever they are disjoint. The kth multichromatic number of the graph G is the least integer t such that the vertices of G can be assigned k-subsets of {1,2, …, t}, so that adjacent vertices of G receive disjoint sets. The values of Xk(K(m,n)) are computed for n = 2, 3 and bounded for n ⩾ 4

Approximating the Orthogonality Dimension of Graphs and Hypergraphs

A t-dimensional orthogonal representation of a hypergraph is an assignment of nonzero vectors in R^t to its vertices, such that every hyperedge contains two vertices whose vectors are orthogonal. The orthogonality dimension of a hypergraph H, denoted by overline{xi}(H), is the smallest integer t for which there exists a t-dimensional orthogonal representation of H. In this paper we study computational aspects of the orthogonality dimension of graphs and hypergraphs. We prove that for every k >= 4, it is NP-hard (resp. quasi-NP-hard) to distinguish n-vertex k-uniform hypergraphs H with overline{xi}(H) = Omega(log^delta n) for some constant delta>0 (resp. overline{xi}(H) >= Omega(log^{1-o(1)} n)). For graphs, we relate the NP-hardness of approximating the orthogonality dimension to a variant of a long-standing conjecture of Stahl. We also consider the algorithmic problem in which given a graph G with overline{xi}(G) <= 3 the goal is to find an orthogonal representation of G of as low dimension as possible, and provide a polynomial time approximation algorithm based on semidefinite programming

The (Generalized) Orthogonality Dimension of (Generalized) Kneser Graphs: Bounds and Applications

The orthogonality dimension of a graph $G=(V,E)$ over a field $\mathbb{F}$ is the smallest integer $t$ for which there exists an assignment of a vector $u_v \in \mathbb{F}^t$ with $\langle u_v,u_v \rangle \neq 0$ to every vertex $v \in V$, such that $\langle u_v, u_{v'} \rangle = 0$ whenever $v$ and $v'$ are adjacent vertices in $G$. The study of the orthogonality dimension of graphs is motivated by various application in information theory and in theoretical computer science. The contribution of the present work is two-folded. First, we prove that there exists a constant $c$ such that for every sufficiently large integer $t$, it is $\mathsf{NP}$-hard to decide whether the orthogonality dimension of an input graph over $\mathbb{R}$ is at most $t$ or at least $3t/2-c$. At the heart of the proof lies a geometric result, which might be of independent interest, on a generalization of the orthogonality dimension parameter for the family of Kneser graphs, analogously to a long-standing conjecture of Stahl (J. Comb. Theo. Ser. B, 1976). Second, we study the smallest possible orthogonality dimension over finite fields of the complement of graphs that do not contain certain fixed subgraphs. In particular, we provide an explicit construction of triangle-free $n$-vertex graphs whose complement has orthogonality dimension over the binary field at most $n^{1-\delta}$ for some constant $\delta >0$. Our results involve constructions from the family of generalized Kneser graphs and they are motivated by the rigidity approach to circuit lower bounds. We use them to answer a couple of questions raised by Codenotti, Pudl\'{a}k, and Resta (Theor. Comput. Sci., 2000), and in particular, to disprove their Odd Alternating Cycle Conjecture over every finite field.Comment: 19 page

k-tuple colorings of the Cartesian product of graphs

A k-tuple coloring of a graph G assigns a set of k colors to each vertex of G such that if two vertices are adjacent, the corresponding sets of colors are disjoint. The k-tuple chromatic number of G, χk(G), is the smallest t so that there is a k-tuple coloring of G using t colors. It is well known that χ(G□H)=max{χ(G),χ(H)}. In this paper, we show that there exist graphs G and H such that χk(G□H)>max{χk(G),χk(H)} for k≥2. Moreover, we also show that there exist graph families such that, for any k≥1, the k-tuple chromatic number of their Cartesian product is equal to the maximum k-tuple chromatic number of its factors.Fil: Bonomo, Flavia. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigación en Ciencias de la Computación. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigación en Ciencias de la Computación; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Computación; ArgentinaFil: Koch, Ivo Valerio. Universidad Nacional de General Sarmiento. Instituto de Industria; ArgentinaFil: Torres, Pablo. Universidad Nacional de Rosario. Facultad de Ciencias Exactas, Ingeniería y Agrimensura; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; ArgentinaFil: Valencia Pabon, Mario. Universite de Paris 13-Nord; Francia. Centre National de la Recherche Scientifique; Franci

On Colorings of Graph Powers

In this paper, some results concerning the colorings of graph powers are presented. The notion of helical graphs is introduced. We show that such graphs are hom-universal with respect to high odd-girth graphs whose $(2t+1)$st power is bounded by a Kneser graph. Also, we consider the problem of existence of homomorphism to odd cycles. We prove that such homomorphism to a $(2k+1)$-cycle exists if and only if the chromatic number of the $(2k+1)$st power of $S_2(G)$ is less than or equal to 3, where $S_2(G)$ is the 2-subdivision of $G$. We also consider Ne\v{s}et\v{r}il's Pentagon problem. This problem is about the existence of high girth cubic graphs which are not homomorphic to the cycle of size five. Several problems which are closely related to Ne\v{s}et\v{r}il's problem are introduced and their relations are presented