Small 1-defective Ramsey numbers in perfect graphs

Abstract

In this paper, we initiate the study of defective Ramsey numbers for the class of perfect graphs. Let $\mathcal{PG}$ be the class of all perfect graphs and $R_1^{\mathcal{PG}}(i,j)$ denote the smallest $n$ such that all perfect graphs on $n$ vertices have either a 1-dense $i$-set or a 1-sparse $j$-set. We show that $R_1^{\mathcal{PG}}(3,j)=j$ for any $j\ge 2$, $R_1^{\mathcal{PG}}(4,4)=6$, $R_1^{\mathcal{PG}}(4,5)=8$, $R_1^{\mathcal{PG}}(4,6)=10$, $R_1^{\mathcal{PG}}(4,7)=13$, $R_1^{\mathcal{PG}}(4,8)=15$ and $R_1^{\mathcal{PG}}(5,5)=13$. We exhibit all extremal graphs for $R_1^{\mathcal{PG}}(4,7)=13$ (there are exactly three). We also obtain the 1-defective Ramsey number of order (4,7) in triangle-free perfect graphs, namely $R_1^{\Delta \mathcal{PG}}(4,7)=12$.

Introduction

For any two positive integers $i$ and $j$, the Ramsey number $R(i,j)$ is the smallest positive integer such that every graph on at least $R(i,j)$ vertices has a clique of size $i$ or an independent set of size $j$. Ramsey numbers along with several variations are extensively studied in the literature. Among various generalizations of the classical Ramsey numbers, we note that defective Ramsey numbers have been the focus of several research papers [1], [2], [3], [4]. This variation is based on the following relaxation of the notions of cliques and independent sets. A $k$-sparse$i$-set is a set $S$ of $i$ vertices of a graph $G$ such that each vertex in $S$ has degree at most $k$ in the subgraph of $G$ induced by $S$. A $k$-dense$j$-set is a set $D$ of $j$ vertices of a graph $G$ that is $k$-sparse in the complement of $G$; in other words, each vertex in $D$ misses at most $k$ other vertices in $D$. The term $k$-defective (or $k$-uniform) set is used to denote a $k$-sparse or $k$-dense set. The defective Ramsey number $R_k^{\mathcal{G}}(i,j)$ is the smallest $n$ such that all graphs on $n$ vertices in the class $\mathcal{G}$ have either a $k$-dense $i$-set or a $k$-sparse $j$-set. In [3] some 1-defective Ramsey numbers are reported under the name of 1-dependent Ramsey numbers. In [2], [4] more 1-defective and 2-defective Ramsey numbers are found using direct proof techniques and several bounds are derived on general defective Ramsey numbers. However, direct proof techniques seem to have reached their limits in finding new values of defective Ramsey numbers. Indeed, this is rather not surprising given the great difficulty of computing specific Ramsey numbers (of all kinds). Having noticed this fact, some computer based generation methods are used in [1] and [2] to improve the known bounds on defective Ramsey numbers (and some other defective parameters).

Noting that computing all Ramsey numbers is notoriously unlikely, various approaches have been adopted in the literature to cope with this difficulty. One way to attack Ramsey numbers is to consider restricted graph families. In [5], all Ramsey numbers in planar graphs are computed. In [6], [7], [8], the authors achieved with the computation of some Ramsey numbers for graphs with bounded degree. Exact values as well as upper and lower bounds on some Ramsey numbers for claw-free graphs are proved in [9]. This approach has gained some importance more recently and become popular again. In [10], the authors conduct a systematic study of Ramsey numbers in various graph classes. They determine some classes of graphs such as claw-free graphs for which there are infinitely many non-trivial Ramsey numbers as for arbitrary graphs. Besides, they exhibit all (classical) Ramsey numbers for perfect graphs and some well-known subclasses of claw-free graphs. Similarly, a very recent work [11] focuses on the complexity of the coloring problem where every color class is a $k$-sparse set (called the defective coloring problem) when restricted to subclasses of perfect graphs.

In this paper, we adopt this natural approach of considering hard-to-compute parameters in special cases. We initiate the study of defective Ramsey numbers for perfect graphs and compute first values for small numbers. Namely, we show that $R_1^{\mathcal{PG}}(3,j)=j$ for any $j\ge 2$, $R_1^{\mathcal{PG}}(4,4)=6$, $R_1^{\mathcal{PG}}(4,5)=8$, $R_1^{\mathcal{PG}}(4,6)=10$, $R_1^{\mathcal{PG}}(4,7)=13$, $R_1^{\mathcal{PG}}(4,8)=15$ and $R_1^{\mathcal{PG}}(5,5)=13$ where $\mathcal{PG}$ denotes the class of perfect graphs. We also exhibit all (there are exactly three) extremal graphs for $R_1^{\mathcal{PG}}(4,7)=13$. To find these extremal graphs, we also obtain (as a byproduct) $R_1^{\Delta \mathcal{PG}}(4,7)=12$ where $\Delta \mathcal{PG}$ denotes the class of triangle-free perfect graphs.

Our main proofs do not use any previously known results. They are based on techniques from classical Ramsey theory such as considering the neighborhood of specific vertices and on the structural properties of perfect graphs and 1-defective sets. We also invoke a computer assisted proof for one of these values. We conclude with several research directions arising from our work.

Let $G=(V,E)$ be a graph. A (partial) subgraph $H\subseteq G$ is a graph on $V^{\prime }\subseteq V$ and $E^{\prime }\subseteq E$ with both end-vertices of each edge of $E^{\prime }$ in $V^{\prime }$. If all edges with both end-vertices in $V^{\prime }$ are in $E^{\prime }$, then $H$ is said to be an induced subgraph of $G$. In our context, whenever we say that a graph contains a subgraph, we always mean as a partial subgraph, unless stated otherwise. For a vertex $x\in V$, we denote by $N\left(x\right)$ the set of neighbors of $x$, that is, vertices adjacent to $x$. The degree of a vertex $x$ is $d\left(x\right)=\left|N\left(x\right)\right|$. We also have $N\left[x\right]=N\left(x\right)\cup \left\{x\right\}$. For a vertex $x\in V$ and a subgraph $H\subseteq G$, we denote by $N_H\left(x\right)$ the set of neighbors of $x$ in $H$, that is $N\left(x\right)\cap V\left(H\right)$. Similarly, the degree of $x$ in $H$ is $d_H\left(x\right)=\left|N_H\left(x\right)\right|$. For a subset of vertices $X\subset V$, the neighborhood of $X$, denoted by $N\left(X\right)$, is defined as $N\left(X\right)=\left({\cup }_{x\in X}N\left(x\right)\right)\setminus X$. For a subset of vertices $X\subset V$ and a subgraph $H\subseteq G$, we define $N_H\left(X\right)$ similarly to $N_H\left(x\right)$.

For a graph $G$ and a subgraph $H$, we use the notation $G\setminus H$ to mean the subgraph of $G$ induced by all vertices in $V\left(G\right)\setminus V\left(H\right)$. We also use the same notation when we remove a set of vertices from a graph. For a graph $H$ and $G$, we say that $G$ is $H$-free if it does not contain $H$ as an induced subgraph. A path on $n$ vertices is denoted by $P_n$, and a cycle on $n$ vertices, also called an $n$-cycle, is denoted by $C_n$. The distance between two vertices is the length of a shortest path between them. The girth of a graph $G$, denoted by $g\left(G\right)$ is the length of a shortest induced cycle in it. A clique of size $i$, denoted by $K_i$, is a set of $i$ vertices which are pairwise adjacent. Given a graph $G$, the maximum size of a clique is denoted by $\omega \left(G\right)$. A set of vertices is called independent if all vertices in it are pairwise non-adjacent.

The chromatic number of $G$, denoted by $\chi \left(G\right)$ is the minimum number of colors to color the vertices of it in such a way that no two adjacent vertices receive the same color. A graph $G$ is called perfect if for every induced subgraph $H$ of $G$, we have $\chi \left(H\right)=\omega \left(H\right)$. According to the Strong Perfect Graph Theorem, a graph is perfect if and only if it contains no induced odd cycles of length at least 5 and their complements [12]. We will be using this result heavily in our proofs. An extremal graph for $R_k^{\mathcal{G}}(i,j)$ is a graph in $\mathcal{G}$ on $R_k^{\mathcal{G}}(i,j)-1$ vertices containing neither a $k$-dense $i$-set nor a $k$-sparse $j$-set. We omit the superscript $\mathcal{G}$ when we consider a Ramsey number in arbitrary graphs.