Chromatic Vertex Folkman Numbers

For graph G and integers a1 \u3e · · · \u3e ar \u3e 2, we write G → (a1, · · · , ar) v if and only if for every r-coloring of the vertex set V (G) there exists a monochromatic Kai in G for some color i ∈ {1, · · · , r}. The vertex Folkman number Fv(a1, · · · , ar; s) is defined as the smallest integer n for which there exists a Ks-free graph G of order n such that G → (a1, · · · , ar) v . It is well known that if G → (a1, · · · , ar) v then χ(G) \u3e m, where m = 1+Pr i=1(ai−1). In this paper we study such Folkman graphs G with chromatic number χ(G) = m, which leads to a new concept of chromatic Folkman numbers. We prove constructively some existential results, among others that for all r, s \u3e 2 there exist Ks+1-free graphs G such that G → (s, · · ·r , s) v and G has the smallest possible chromatic number r(s − 1) + 1 with respect to this property. Among others we conjecture that for every s \u3e 2 there exists a Ks+1-free graph G on Fv(s, s; s + 1) vertices with χ(G) = 2s − 1 and G → (s, s) v

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

Solving Hard Graph Problems with Combinatorial Computing and Optimization

Many problems arising in graph theory are difficult by nature, and finding solutions to large or complex instances of them often require the use of computers. As some such problems are $NP$-hard or lie even higher in the polynomial hierarchy, it is unlikely that efficient, exact algorithms will solve them. Therefore, alternative computational methods are used. Combinatorial computing is a branch of mathematics and computer science concerned with these methods, where algorithms are developed to generate and search through combinatorial structures in order to determine certain properties of them. In this thesis, we explore a number of such techniques, in the hopes of solving specific problem instances of interest. Three separate problems are considered, each of which is attacked with different methods of combinatorial computing and optimization. The first, originally proposed by ErdH{o}s and Hajnal in 1967, asks to find the Folkman number $F_e(3,3;4)$, defined as the smallest order of a $K_4$-free graph that is not the union of two triangle-free graphs. A notoriously difficult problem associated with Ramsey theory, the best known bounds on it prior to this work were $19 leq F_e(3,3;4) leq 941$. We improve the upper bound to $F_e(3,3;4) leq 786$ using a combination of known methods and the Goemans-Williamson semi-definite programming relaxation of MAX-CUT. The second problem of interest is the Ramsey number $R(C_4,K_m)$, which is the smallest $n$ such that any $n$-vertex graph contains a cycle of length four or an independent set of order $m$. With the help of combinatorial algorithms, we determine $R(C_4,K_9)=30$ and $R(C_4,K_{10})=36$ using large-scale computations on the Open Science Grid. Finally, we explore applications of the well-known Lenstra-Lenstra-Lov\u27{a}sz (LLL) algorithm, a polynomial-time algorithm that, when given a basis of a lattice, returns a basis for the same lattice with relatively short vectors. The main result of this work is an application to graph domination, where certain hard instances are solved using this algorithm as a heuristic