Complexity of stability

We investigate the notion of quantum chromatic number of a graph, which is the minimal number of colours necessary in a protocol in which two separated provers can convince a referee that they have a colouring of the graph.After discussing this notion from first principles, we go on to establish relations with the clique number and orthogonal representations of the graph. We also prove several general facts about this graph parameter and find large separations between the clique number and the quantum chromatic number by looking at random graphs. Finally, we show that there can be no separation between classical and quantum chromatic number if the latter is $2$, nor if it is $3$ in a restricted quantum model; on the other hand, we exhibit a graph on $18$ vertices and $44$ edges with chromatic number $5$ and quantum chromatic number $4$.

Maximizing the degree powers of graphs with fixed size

We prove bounds on the chromatic number $\chi$ of a vertex-transitive graph in terms of its clique number $\omega$ and maximum degree $\Delta$. We conjecture that every vertex-transitive graph satisfies $\chi \le \max \{\omega, \left\lceil\frac{5\Delta + 3}{6}\right\rceil\}$, and we prove results supporting this conjecture. Finally, for vertex-transitive graphs with $\Delta \ge 13$ we prove the Borodin–Kostochka conjecture, i.e., $\chi\le\max\{\omega,\Delta-1\}$.

For a hypergraph H on the vertex set {1,...,n}, a distribution D = (D_1,...,D_n) over {0,1}^n is H-wise independent if every restriction of D to indices which form an edge in H is uniform. This generalizes the notion of k-wise independence obtained by taking H to be the complete n vertex k-uniform hypergraph. This generalization was studied by Schulman (STOC 1992), who presented constructions of H-wise independent distributions that are linear, i.e., the samples are strings of inner products (over F2) of a fixed set of vectors with a uniformly chosen random vector. Let l(H) denote the minimum possible size of a sample space of a uniform H-wise independent distribution. The l parameter is well understood for the special case of k-wise independence. In this work we study the notion of H-wise independence and the l parameter for general graphs and hypergraphs. For graphs, we show how the l parameter relates to standard graph parameters (e.g., clique number, chromatic number, Lovasz theta function, minrank). We derive algorithmic and hardness results for this parameter as well as an explicit construction of graphs G for which l(G) is exponentially smaller than the size of the sample space of any linear G-wise independent distribution. For hypergraphs, we study the problem of testing whether a given distribution is H-wise independent, generalizing results of Alon et al. (STOC 2007).

Nordhaus–Gaddum inequalities for the fractional and circular chromatic numbers

For a nontrivial connected graph $G$ of order $n$, the detour distance $D(u,v)$ between two vertices $u$ and $v$ in $G$ is the length of a longest $u-v$ path in $G$. Detour distance is a metric on the vertex set of $G$. For each integer $k$ with $1\le k\le n-1$, a coloring $c\colon V(G)\to \mathbb N$ is a $k$-metric coloring of $G$ if $|c(u)-c(v)|+D(u,v)\ge k+1$ for every two distinct vertices $u$ and $v$ of $G$. The value $\chi _m^k(c)$ of a $k$-metric coloring $c$ is the maximum color assigned by $c$ to a vertex of $G$ and the $k$-metric chromatic number $\chi _m^k(G)$ of $G$ is the minimum value of a $k$-metric coloring of $G$. For every nontrivial connected graph $G$ of order $n$, $\chi _m^1(G)\le \chi _m^2(G)\le \cdots \le \chi _m^{n-1}(G)$. Metric chromatic numbers provide a generalization of several well-studied coloring parameters in graphs. Upper and lower bounds have been established for $\chi _m^k(G)$ in terms of other graphical parameters of a graph $G$ and exact values of $k$-metric chromatic numbers have been determined for complete multipartite graphs and cycles. For a nontrivial connected graph $G$, the anti-diameter ${\rm adiam}(G)$ is the minimum detour distance between two vertices of $G$. We show that the ${\rm adiam}(G)$-metric chromatic number of a graph $G$ provides information on the Hamiltonian properties of the graph and investigate realization results and problems on this parameter.

The reciprocal distance Laplacian matrix of a connected graph G is defined as RDL(G)=RT(G)−RD(G), where RT(G) is the diagonal matrix of reciprocal distance degrees and RD(G) is the Harary matrix. Clearly, RDL(G) is a real symmetric matrix, and we denote its eigenvalues as λ1(RDL(G))≥λ2(RDL(G))≥…≥λn(RDL(G)). The largest eigenvalue λ1(RDL(G)) of RDL(G), denoted by λ(G), is called the reciprocal distance Laplacian spectral radius. In this paper, we obtain several upper bounds for the sum of k largest reciprocal distance Laplacian eigenvalues of G in terms of various graph parameters, such as order n, maximum reciprocal distance degree RTmax, minimum reciprocal distance degree RTmin, and Harary index H(G) of G. We determine the extremal cases corresponding to these bounds. As a consequence, we obtain the upper bounds for reciprocal distance Laplacian spectral radius λ(G) in terms of the parameters as mentioned above and characterize the extremal cases. Moreover, we attain several upper and lower bounds for reciprocal distance Laplacian spread RDLS(G)=λ1(RDL(G))−λn−1(RDL(G)) in terms of various graph parameters. We determine the extremal graphs in many cases.

We say a graph has property $\mathcal{P}_{g,p}$ when it is an induced subgraph of the curve graph of a surface of genus $g$ with $p$ punctures. Two well-known graph invariants, the chromatic and clique numbers, can provide obstructions to $\mathcal{P}_{g,p}$. We introduce a new invariant of a graph, the \emph{nested complexity length}, which provides a novel obstruction to $\mathcal{P}_{g,p}$. For the curve graph this invariant captures the topological complexity of the surface in graph-theoretic terms; indeed we show that its value is $6g-6+2p$, i.e. twice the size of a maximal multicurve on the surface. As a consequence we show that large `half-graphs' do not have $\mathcal{P}_{g,p}$, and we deduce quantitatively that almost all finite graphs which pass the chromatic and clique tests do not have $\mathcal{P}_{g,p}$. We also reinterpret our obstruction in terms of the first-order theory of the curve graph, and in terms of RAAG subgroups of the mapping class group (following Kim and Koberda). Finally, we show that large complete multipartite graphs cannot have $\mathcal{P}_{g,p}$. This allows us to compute the upper density of the curve graph, and to conclude that clique size, chromatic number, and nested complexity length are not sufficient to determine $\mathcal{P}_{g,p}$.

We investigate hierarchies of semidefinite approximations for the chromatic number $\chi(G)$ of a graph G. We introduce an operator $\Psi$ mapping any graph parameter $\beta(G)$, nested between the stability number $\alpha(G)$ and $\chi(\overline G)$, to a new graph parameter $\Psi_\beta(G)$, nested between $\alpha (\overline G)$ and $\chi(G)$; $\Psi_\beta(G)$ is polynomial time computable if $\beta(G)$ is. As an application, there is no polynomial time computable graph parameter nested between the fractional chromatic number $\chi^*(\cdot)$ and $\chi(\cdot)$ unless $\text{P} = \text{NP}$. Moreover, based on the Motzkin–Straus formulation for $\alpha(G)$, we give (quadratically constrained) quadratic and copositive programming formulations for $\chi(G)$. Under some mild assumptions, $n/\beta(G)\le \Psi_\beta(G)$, but, while $n/\beta(G)$ remains below $\chi^*(G)$, $\Psi_\beta(G)$ can reach $\chi(G)$ (e.g., for $\beta(\cdot)=\alpha(\cdot)$). We also define new polynomial time computable lower bounds for $\chi(G)$, improving the classic Lovász theta number (and its strengthenings obtained by adding nonnegativity and triangle inequalities); experimental results on Hamming graphs, Kneser graphs, and DIMACS benchmark graphs will be given in the follow-up paper [N. Gvozdenović and M. Laurent, SIAM J. Optim., 19 (2008), pp. 592–615].

Circular chromatic numbers of Mycielski's graphs

Some Theorems Concerning the Star Chromatic Number of a Graph

Coloring the Vertices of a Graph With Measurable Sets in a Probability Space

In this paper we give upper bounds on the number of minimal separators and potential maximal cliques of graphs w.r.t. two graph parameters, namely vertex cover (vc) and modular width (mw). We prove that for any graph, the number of minimal separators is \(\mathcal{O}^*(3^{\operatorname{vc}})\) and \(\mathcal{O}^*(1.6181^{\operatorname{mw}})\), the number of potential maximal cliques is \(\mathcal{O}^*(4^{\operatorname{vc}})\) and \(\mathcal{O}^*(1.7347^{\operatorname{mw}})\), and these objects can be listed within the same running times. (The \(\mathcal{O}^*\) notation suppresses polynomial factors in the size of the input.) Combined with known results [3,12], we deduce that a large family of problems, e.g., Treewidth, Minimum Fill-in, Longest Induced Path, Feedback vertex set and many others, can be solved in time \(\mathcal{O}^*(4^{\operatorname{vc}})\) or \(\mathcal{O}^*(1.7347^{\operatorname{mw}})\).

In this paper, we investigate some basic properties of fractional powers. In this regard, we show that for any non-bipartite graph $G$ and positive rational numbers ${2r+1\over 2s+1} < {2p+1\over 2q+1}$, we have $G^{2r+1\over 2s+1} < G^{2p+1\over 2q+1}$. Next, we study the power thickness of $G$, that is, the supremum of rational numbers ${2r+1\over 2s+1}$ such that $G$ and $G^{2r+1\over 2s+1}$ have the same chromatic number. We prove that the power thickness of any non-complete circular complete graph is greater than one. This provides a sufficient condition for the equality of the chromatic number and the circular chromatic number of graphs. Finally, we introduce an equivalent definition for the circular chromatic number of graphs in terms of fractional powers. Also, we show that for any non-bipartite graph $G$ if $0 < {{2r+1}\over {2s+1}} \leq {{\chi(G)}\over{3(\chi(G)-2)}}$, then $\chi(G^{{2r+1}\over {2s+1}})=3$. Moreover, $\chi(G)\neq\chi_c(G)$ if and only if there exists a rational number ${{2r+1}\over {2s+1}}>{{\chi(G)}\over{3(\chi(G)-2)}}$ for which $\chi(G^{{2r+1}\over {2s+1}})= 3$.

A k-decomposition (G1,…,Gk) of a graph G is a partition of its edge set to form k spanning subgraphs G1,…,Gk. The classical theorem of Nordhaus and Gaddum bounds χ(G1) + χ(G2) and χ(G1)χ(G2) over all 2-decompositions of Kn. For a graph parameter p, let p(k;G) denote the maximum of $\sum _{i=1}^k p(G_i)$ over all k-decompositions of the graph G. The clique number ω, chromatic number χ, list chromatic number χ, and SzekeresWilf number σ satisfy ω(2;Kn) = χ(2;Kn) = χl(2;Kn) = σ(2;Kn) = n + 1. We obtain lower and upper bounds for ω(k;Kn), χ(k;Kn), χl(k;Kn), and σ(k;Kn). The last three behave differently for large k. We also obtain lower and upper bounds for the maximum of χ(k;G) over all graphs embedded on a given surface. © 2005 Wiley Periodicals, Inc. J Graph Theory