Irregular Colorings of Regular Graphs

An irregular coloring of a graph is a proper vertex coloring that distinguishes vertices in the graph either by their own colors or by the colors of their neighbors. In algebraic graph theory, graphs with a certain amount of symmetry can sometimes be specified in terms of a group and a smaller graph called a voltage graph. Radcliffe and Zhang found a bound for the irregular chromatic number of a graph on n vertices. In this paper we use voltage graphs to construct graphs achieving that bound

A reverse Sidorenko inequality

Let $H$ be a graph allowing loops as well as vertex and edge weights. We prove that, for every triangle-free graph $G$ without isolated vertices, the weighted number of graph homomorphisms $\hom(G, H)$ satisfies the inequality $$\hom(G, H ) \le \prod_{uv \in E(G)} \hom(K_{d_u,d_v}, H )^{1/(d_ud_v)},$$ where $d_u$ denotes the degree of vertex $u$ in $G$. In particular, one has $$\hom(G, H )^{1/|E(G)|} \le \hom(K_{d,d}, H )^{1/d^2}$$ for every $d$-regular triangle-free $G$. The triangle-free hypothesis on $G$ is best possible. More generally, we prove a graphical Brascamp-Lieb type inequality, where every edge of $G$ is assigned some two-variable function. These inequalities imply tight upper bounds on the partition function of various statistical models such as the Ising and Potts models, which includes independent sets and graph colorings. For graph colorings, corresponding to $H = K_q$, we show that the triangle-free hypothesis on $G$ may be dropped; this is also valid if some of the vertices of $K_q$ are looped. A corollary is that among $d$-regular graphs, $G = K_{d,d}$ maximizes the quantity $c_q(G)^{1/|V(G)|}$ for every $q$ and $d$, where $c_q(G)$ counts proper $q$-colorings of $G$. Finally, we show that if the edge-weight matrix of $H$ is positive semidefinite, then $$\hom(G, H) \le \prod_{v \in V(G)} \hom(K_{d_v+1}, H )^{1/(d_v+1)}.$$ This implies that among $d$-regular graphs, $G = K_{d+1}$ maximizes $\hom(G, H)^{1/|V(G)|}$. For 2-spin Ising models, our results give a complete characterization of extremal graphs: complete bipartite graphs maximize the partition function of 2-spin antiferromagnetic models and cliques maximize the partition function of ferromagnetic models. These results settle a number of conjectures by Galvin-Tetali, Galvin, and Cohen-Csikv\'ari-Perkins-Tetali, and provide an alternate proof to a conjecture by Kahn.Comment: 30 page

Vertex-Coloring 2-Edge-Weighting of Graphs

A $k$-{\it edge-weighting} $w$ of a graph $G$ is an assignment of an integer weight, $w(e)\in \{1,\dots, k\}$, to each edge $e$. An edge weighting naturally induces a vertex coloring $c$ by defining $c(u)=\sum_{u\sim e} w(e)$ for every $u \in V(G)$. A $k$-edge-weighting of a graph $G$ is \emph{vertex-coloring} if the induced coloring $c$ is proper, i.e., $c(u) \neq c(v)$ for any edge $uv \in E(G)$. Given a graph $G$ and a vertex coloring $c_0$, does there exist an edge-weighting such that the induced vertex coloring is $c_0$? We investigate this problem by considering edge-weightings defined on an abelian group. It was proved that every 3-colorable graph admits a vertex-coloring $3$-edge-weighting \cite{KLT}. Does every 2-colorable graph (i.e., bipartite graphs) admit a vertex-coloring 2-edge-weighting? We obtain several simple sufficient conditions for graphs to be vertex-coloring 2-edge-weighting. In particular, we show that 3-connected bipartite graphs admit vertex-coloring 2-edge-weighting

Group twin coloring of graphs

For a given graph $G$, the least integer $k\geq 2$ such that for every Abelian group $\mathcal{G}$ of order $k$ there exists a proper edge labeling $f:E(G)\rightarrow \mathcal{G}$ so that $\sum_{x\in N(u)}f(xu)\neq \sum_{x\in N(v)}f(xv)$ for each edge $uv\in E(G)$ is called the \textit{group twin chromatic index} of $G$ and denoted by $\chi'_g(G)$. This graph invariant is related to a few well-known problems in the field of neighbor distinguishing graph colorings. We conjecture that $\chi'_g(G)\leq \Delta(G)+3$ for all graphs without isolated edges, where $\Delta(G)$ is the maximum degree of $G$, and provide an infinite family of connected graph (trees) for which the equality holds. We prove that this conjecture is valid for all trees, and then apply this result as the base case for proving a general upper bound for all graphs $G$ without isolated edges: $\chi'_g(G)\leq 2(\Delta(G)+{\rm col}(G))-5$, where ${\rm col}(G)$ denotes the coloring number of $G$. This improves the best known upper bound known previously only for the case of cyclic groups $\mathbb{Z}_k$

On two problems in Ramsey-Tur\'an theory

Alon, Balogh, Keevash and Sudakov proved that the $(k-1)$-partite Tur\'an graph maximizes the number of distinct $r$-edge-colorings with no monochromatic $K_k$ for all fixed $k$ and $r=2,3$, among all $n$-vertex graphs. In this paper, we determine this function asymptotically for $r=2$ among $n$-vertex graphs with sub-linear independence number. Somewhat surprisingly, unlike Alon-Balogh-Keevash-Sudakov's result, the extremal construction from Ramsey-Tur\'an theory, as a natural candidate, does not maximize the number of distinct edge-colorings with no monochromatic cliques among all graphs with sub-linear independence number, even in the 2-colored case. In the second problem, we determine the maximum number of triangles asymptotically in an $n$-vertex $K_k$-free graph $G$ with $\alpha(G)=o(n)$. The extremal graphs have similar structure to the extremal graphs for the classical Ramsey-Tur\'an problem, i.e.~when the number of edges is maximized.Comment: 22 page