On Some Generalized Vertex Folkman Numbers

For a graph $G$ and integers $a_i\ge 2$, the expression $G \rightarrow (a_1,\dots,a_r)^v$ means that for any $r$-coloring of the vertices of $G$ there exists a monochromatic $a_i$-clique in $G$ for some color $i \in \{1,\cdots,r\}$. The vertex Folkman numbers are defined as $F_v(a_1,\dots,a_r;H) = \min\{|V(G)| : G$ is $H$-free and $G \rightarrow (a_1,\dots,a_r)^v\}$, where $H$ is a graph. Such vertex Folkman numbers have been extensively studied for $H=K_s$ with $s>\max\{a_i\}_{1\le i \le r}$. If $a_i=a$ for all $i$, then we use notation $F_v(a^r;H)=F_v(a_1,\dots,a_r;H)$. Let $J_k$ be the complete graph $K_k$ missing one edge, i.e. $J_k=K_k-e$. In this work we focus on vertex Folkman numbers with $H=J_k$, in particular for $k=4$ and $a_i\le 3$. A result by Ne\v{s}et\v{r}il and R\"{o}dl from 1976 implies that $F_v(3^r;J_4)$ is well defined for any $r\ge 2$. We present a new and more direct proof of this fact. The simplest but already intriguing case is that of $F_v(3,3;J_4)$, for which we establish the upper bound of 135. We obtain the exact values and bounds for a few other small cases of $F_v(a_1,\dots,a_r;J_4)$ when $a_i \le 3$ for all $1 \le i \le r$, including $F_v(2,3;J_4)=14$, $F_v(2^4;J_4)=15$, and $22 \le F_v(2^5;J_4) \le 25$. Note that $F_v(2^r;J_4)$ is the smallest number of vertices in any $J_4$-free graph with chromatic number $r+1$. Most of the results were obtained with the help of computations, but some of the upper bound graphs we found are interesting by themselves

Circular Coloring of Random Graphs: Statistical Physics Investigation

Circular coloring is a constraints satisfaction problem where colors are assigned to nodes in a graph in such a way that every pair of connected nodes has two consecutive colors (the first color being consecutive to the last). We study circular coloring of random graphs using the cavity method. We identify two very interesting properties of this problem. For sufficiently many color and sufficiently low temperature there is a spontaneous breaking of the circular symmetry between colors and a phase transition forwards a ferromagnet-like phase. Our second main result concerns 5-circular coloring of random 3-regular graphs. While this case is found colorable, we conclude that the description via one-step replica symmetry breaking is not sufficient. We observe that simulated annealing is very efficient to find proper colorings for this case. The 5-circular coloring of 3-regular random graphs thus provides a first known example of a problem where the ground state energy is known to be exactly zero yet the space of solutions probably requires a full-step replica symmetry breaking treatment.Comment: 19 pages, 8 figures, 3 table

Data Reduction for Graph Coloring Problems

This paper studies the kernelization complexity of graph coloring problems with respect to certain structural parameterizations of the input instances. We are interested in how well polynomial-time data reduction can provably shrink instances of coloring problems, in terms of the chosen parameter. It is well known that deciding 3-colorability is already NP-complete, hence parameterizing by the requested number of colors is not fruitful. Instead, we pick up on a research thread initiated by Cai (DAM, 2003) who studied coloring problems parameterized by the modification distance of the input graph to a graph class on which coloring is polynomial-time solvable; for example parameterizing by the number k of vertex-deletions needed to make the graph chordal. We obtain various upper and lower bounds for kernels of such parameterizations of q-Coloring, complementing Cai's study of the time complexity with respect to these parameters. Our results show that the existence of polynomial kernels for q-Coloring parameterized by the vertex-deletion distance to a graph class F is strongly related to the existence of a function f(q) which bounds the number of vertices which are needed to preserve the NO-answer to an instance of q-List-Coloring on F.Comment: Author-accepted manuscript of the article that will appear in the FCT 2011 special issue of Information & Computatio

Complexity of C_k-Coloring in Hereditary Classes of Graphs

For a graph F, a graph G is F-free if it does not contain an induced subgraph isomorphic to F. For two graphs G and H, an H-coloring of G is a mapping f:V(G) -> V(H) such that for every edge uv in E(G) it holds that f(u)f(v)in E(H). We are interested in the complexity of the problem H-Coloring, which asks for the existence of an H-coloring of an input graph G. In particular, we consider H-Coloring of F-free graphs, where F is a fixed graph and H is an odd cycle of length at least 5. This problem is closely related to the well known open problem of determining the complexity of 3-Coloring of P_t-free graphs. We show that for every odd k >= 5 the C_k-Coloring problem, even in the precoloring-extension variant, can be solved in polynomial time in P_9-free graphs. On the other hand, we prove that the extension version of C_k-Coloring is NP-complete for F-free graphs whenever some component of F is not a subgraph of a subdivided claw

Graph coloring with no large monochromatic components

For a graph G and an integer t we let mcc_t(G) be the smallest m such that there exists a coloring of the vertices of G by t colors with no monochromatic connected subgraph having more than m vertices. Let F be any nontrivial minor-closed family of graphs. We show that \mcc_2(G) = O(n^{2/3}) for any n-vertex graph G \in F. This bound is asymptotically optimal and it is attained for planar graphs. More generally, for every such F and every fixed t we show that mcc_t(G)=O(n^{2/(t+1)}). On the other hand we have examples of graphs G with no K_{t+3} minor and with mcc_t(G)=\Omega(n^{2/(2t-1)}). It is also interesting to consider graphs of bounded degrees. Haxell, Szabo, and Tardos proved \mcc_2(G) \leq 20000 for every graph G of maximum degree 5. We show that there are n-vertex 7-regular graphs G with \mcc_2(G)=\Omega(n), and more sharply, for every \epsilon>0 there exists c_\epsilon>0 and n-vertex graphs of maximum degree 7, average degree at most 6+\epsilon for all subgraphs, and with mcc_2(G)\ge c_\eps n. For 6-regular graphs it is known only that the maximum order of magnitude of \mcc_2 is between \sqrt n and n. We also offer a Ramsey-theoretic perspective of the quantity \mcc_t(G).Comment: 13 pages, 2 figure