Vector Chromatic Number Research Articles

Coloring the normalized Laplacian for oriented hypergraphs

The independence number, coloring number and related parameters are investigated in the setting of oriented hypergraphs using the spectrum of the normalized Laplace operator. For the independence number, both an inertia–like bound and a ratio–like bound are shown. A Sandwich Theorem involving the clique number, the vector chromatic number and the coloring number is proved, as well as a lower bound for the vector chromatic number in terms of the smallest and the largest eigenvalue of the normalized Laplacian. In addition, spectral partition numbers are studied in relation to the coloring number.

More tales of Hoffman: bounds for the vector chromatic number of a graph

Let $\chi(G)$ denote the chromatic number of a graph and $\chi_v(G)$ denote the vector chromatic number. For all graphs $\chi_v(G) \le \chi(G)$ and for some graphs $\chi_v(G) \ll \chi(G)$. Galtman proved that Hoffman's well-known lower bound for $\chi(G)$ is in fact a lower bound for $\chi_v(G)$. We prove that two more spectral lower bounds for $\chi(G)$ are also lower bounds for $\chi_v(G)$. We then use one of these bounds to derive a new characterization of $\chi_v(G)$.

Graph homomorphisms via vector colorings

In this paper we study the existence of homomorphisms G→H using semidefinite programming. Specifically, we use the vector chromatic number of a graph, defined as the smallest real number t≥2 for which there exists an assignment of unit vectors i↦pi to its vertices such that 〈pi,pj〉≤−1∕(t−1), when i∼j. Our approach allows to reprove, without using the Erdős–Ko–Rado Theorem, that for n>2r the Kneser graph Kn:r and the q-Kneser graph qKn:r are cores, and furthermore, that for n∕r=n′∕r′ there exists a homomorphism Kn:r→Kn′:r′ if and only if n divides n′. In terms of new applications, we show that the even-weight component of the distance k-graph of the n-cube Hn,k is a core and also, that non-bipartite Taylor graphs are cores. Additionally, we give a necessary and sufficient condition for the existence of homomorphisms Hn,k→Hn′,k′ when n∕k=n′∕k′. Lastly, we show that if a 2-walk-regular graph (which is non-bipartite and not complete multipartite) has a unique optimal vector coloring, it is a core. Based on this sufficient condition we conducted a computational study on Ted Spence’s list of strongly regular graphs (http://www.maths.gla.ac.uk/ es/srgraphs.php) and found that at least 84% are cores.

An Inertial Lower Bound for the Chromatic Number of a Graph

Let $\chi(G$) and $\chi_f(G)$ denote the chromatic and fractional chromatic numbers of a graph $G$, and let $(n^+ , n^0 , n^-)$ denote the inertia of $G$. We prove that:\[1 + \max\left(\frac{n^+}{n^-} , \frac{n^-}{n^+}\right) \le \chi(G)\] and conjecture that \[ 1 + \max\left(\frac{n^+}{n^-} , \frac{n^-}{n^+}\right) \le \chi_f(G).\] We investigate extremal graphs for these bounds and demonstrate that this inertial bound is not a lower bound for the vector chromatic number. We conclude with a discussion of asymmetry between $n^+$ and $n^-$, including some Nordhaus-Gaddum bounds for inertia.

Spotting Trees with Few Leaves

We show two results related to finding trees and paths in graphs. First, we show that in $O^*(1.657^k2^{l/2})$ time one can either find a $k$-vertex tree with $l$ leaves in an $n$-vertex undirected graph or conclude that such a tree does not exist. Our solution can be applied as a subroutine to solve the $k$-Internal Spanning Tree problem in $O^*(min(3.455^k, 1.946^n))$ time using polynomial space, improving upon previous algorithms for this problem. In particular, for the first time we break the natural barrier of $O^*(2^n)$. Second, we show that the running time can be improved whenever the host graph admits a vertex coloring with few colors; it can be an ordinary proper vertex coloring, a fractional vertex coloring, or a vector coloring. In effect, we show improved bounds for Hamiltonicity and $k$-Path in any graph of maximum degree $\Delta=4,\ldots,12$ or with vector chromatic number at most 8. Our results extend the technique by Bjorklund [SIAM J. Comput., 43 (2014), pp. 280--299] and Bjorklund et al...

Cubical coloring — fractional covering by cuts and semidefinite programming

We introduce a new graph parameter that measures fractional covering of a graph by cuts. Besides being interesting in its own right, it is useful for study of homomorphisms and tension-continuous mappings. We study the relations with chromatic number, bipartite density, and other graph parameters. We find the value of our parameter for a family of graphs based on hypercubes. These graphs play for our parameter the role that cliques play for the chromatic number and Kneser graphs for the fractional chromatic number. The fact that the defined parameter attains on these graphs the correct value suggests that our definition is a natural one. In the proof we use the eigenvalue bound for maximum cut and a recent result of Engström, Färnqvist, Jonsson, and Thapper [An approximability-related parameter on graphs – properties and applications, DMTCS vol. 17:1, 2015, 33–66]. We also provide a polynomial time approximation algorithm based on semidefinite programming and in particular on vector chromatic number (defined by Karger, Motwani and Sudan [Approximate graph coloring by semidefinite programming, J. ACM 45 (1998), no. 2, 246–265]).

Sabidussi versus Hedetniemi for three variations of the chromatic number

We investigate vector chromatic number (źvec), Lovasz V-function of the complement $$(\bar \vartheta )$$, and quantum chromatic number (źq) from the perspective of graph homomorphisms. We prove an analog of Sabidussi's theorem for each of these parameters, i.e., that for each of the parameters, the value on the Cartesian product of graphs is equal to the maximum of the values on the factors. Interestingly, as a consequence of this result for $$\bar \vartheta$$, we obtain analog of Hedetniemi's conjecture, i.e., that the value of $$\bar \vartheta$$ on the categorical product of graphs is equal to the minimum of its values on the factors. We conjecture that the analogous results hold for vector and quantum chromatic number, and we prove that this is the case for some special classes of graphs.

On hard instances of approximate vertex cover

We show that if there is a 2 - ϵ approximation algorithm for vertex cover on graphs with vector chromatic number at most 2 + δ, then there is a 2 - f (ϵ, δ) approximation algorithm for vertex cover for all graphs.

Graphs with Tiny Vector Chromatic Numbers and Huge Chromatic Numbers

Karger, Motwani, and Sudan [J. ACM, 45 (1998), pp. 246--265] introduced the notion of a vector coloring of a graph. In particular, they showed that every k-colorable graph is also vector k-colorable, and that for constant k, graphs that are vector k-colorable can be colored by roughly $\Delta^{1 - 2/k}$ colors. Here $\Delta$ is the maximum degree in the graph and is assumed to be of the order of $n^{\delta}$ for some $0 < \delta < 1$. Their results play a major role in the best approximation algorithms used for coloring and for maximum independent sets. We show that for every positive integer k there are graphs that are vector k-colorable but do not have independent sets significantly larger than $n/\Delta^{1 - 2/k}$ (and hence cannot be colored with significantly fewer than $\Delta^{1 - 2/k}$ colors). For $k = O(\log n/\log\log n)$ we show vector k-colorable graphs that do not have independent sets of size (log n)c, for some constant c. This shows that the vector chromatic number does not approximate the chromatic number within factors better than n/polylog n. As part of our proof, we analyze "property testing" algorithms that distinguish between graphs that have an independent set of size n/k, and graphs that are "far" from having such an independent set. Our bounds on the sample size improve previous bounds of Goldreich, Goldwasser, and Ron [J. ACM, 45 (1998), pp. 653--750] for this problem.