On semi-transitive orientability of triangle-free graphs

An orientation of a graph is semi-transitive if it is acyclic, and for any directed path either there is no arc between and , or is an arc for all . An undirected graph is semi-transitive if it admits a semi-transitive orientation. Semi-transitive graphs generalize several important classes of graphs and they are precisely the class of word-representable graphs studied extensively in the literature. Determining if a triangle-free graph is semi-transitive is an NP-hard problem. The existence of non-semi-transitive triangle-free graphs was established via Erdős' theorem by Halldórsson and the authors in 2011. However, no explicit examples of such graphs were known until recent work of the first author and Saito who have shown computationally that a certain subgraph on 16 vertices of the triangle-free Kneser graph is not semi-transitive, and have raised the question on the existence of smaller triangle-free non-semi-transitive graphs. In this paper we prove that the smallest triangle-free 4-chromatic graph on 11 vertices (the Gr"otzsch graph) and the smallest triangle-free 4-chromatic 4-regular graph on 12 vertices (the Chvátal graph) are not semi-transitive. Hence, the Gr"otzsch graph is the smallest triangle-free non-semi-transitive graph. We also prove the existence of semi-transitive graphs of girth 4 with chromatic number 4 including a small one (the circulant graph on 13 vertices) and dense ones (Toft's graphs). Finally, we show that each -regular circulant graph (possibly containing triangles) is semi-transitive

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

Bounds for the smallest k-chromatic graphs of given girth

Let n(g)(k) denote the smallest order of a k-chromatic graph of girth at least g. We consider the problem of determining n(g)(k) for small values of k and g. After giving an overview of what is known about n(g)(k), we provide some new lower bounds based on exhaustive searches, and then obtain several new upper bounds using computer algorithms for the construction of witnesses, and for the verification of their correctness. We also present the first examples of reasonably small order for k = 4 and g > 5. In particular, the new bounds include: n(4)(7) <= 77, 26 <= n(6)(4) <= 66 and 30 <= n(7)(4) <= 171

The smallest 5-chromatic tournament

A coloring of a digraph is a partition of its vertex set such that each class induces a digraph with no directed cycles. A digraph is $k$-chromatic if $k$ is the minimum number of classes in such partition, and a digraph is oriented if there is at most one arc between each pair of vertices. Clearly, the smallest $k$-chromatic digraph is the complete digraph on $k$ vertices, but determining the order of the smallest $k$-chromatic oriented graphs is a challenging problem. It is known that the smallest $2$-, $3$- and $4$-chromatic oriented graphs have $3$, $7$ and $11$ vertices, respectively. In 1994, Neumann-Lara conjectured that a smallest $5$-chromatic oriented graph has $17$ vertices. We solve this conjecture and show that the correct order is $19$

Graph Arrowing: Constructions and Complexity

Graph arrowing is concerned with determining which monochromatic subgraphs are unavoidable when coloring a given graph. There are two main avenues of research concerning arrowing: finding extremal Ramsey/Folkman graphs and categorizing the complexity of arrowing problems. Both avenues have been studied extensively for decades. In this thesis, we focus on graph arrowing problems where one of the monochromatic subgraphs being avoided is the path on three vertices, denoted as P3. Our main contributions involve computing Folkman numbers by generating graphs up to 13 vertices and proving the coNP-completeness of some arrowing problems using a novel reduction framework geared towards avoiding P3\u27s. The (P3,H)-Arrowing Problem asks whether a given graph can be colored using two colors (red and blue) such that there are no red P3\u27s and no blue H\u27s, where H is a fixed graph. The few previous hardness proofs for arrowing problems relied on ad-hoc, laborious constructions of gadgets. We introduce a general framework that can be used to prove the coNP-completeness of (P3,H)-arrowing problems. We search for gadgets computationally. These gadgets allow us to simulate variants of SAT, thus showing coNP-hardness. Finally, we use our (P3,H)-Arrowing hardness reductions to gain insight into variants of Monotone SAT. For fixed k in {4,5,6}, we show that Monotone SAT remains NP-complete under the following constraints: 1) each clause consists of exactly two unnegated literals or exactly k negated literals, 2) the variables in each clause are distinct, and 3) the number of times a variable occurs in the formula is bounded by a constant. For future work, we expect that the insight gained by our computationally assisted reductions will help us prove the complexity of other elusive arrowing problems

On minimal triangle‐free 6‐chromatic graphs

A graph with chromatic number k is called k-chromatic. Using computational methods, we show that the smallest triangle-free 6-chromatic graphs have at least 32 and at most 40 vertices. We also determine the complete set of all triangle-free 5-chromatic graphs up to 24 vertices. This implies that Reed's conjecture holds for triangle-free graphs up to at least this order. We also establish that a smallest regular triangle-free 5-chromatic graph has 24 vertices. Finally, we show that the smallest 5-chromatic graphs of girth at least 5 have at least 29 vertices and that the smallest 4-chromatic graphs of girth at least 6 have at least 25 vertices