Finite subgraphs of uncountably chromatic graphs

Theory and Applications of Graphs

Given a vertex coloring c of a graph, the neighborhood color set of a vertex is defined to be the set of all of its neighbors' colors. The coloring c is called a set coloring if any two adjacent vertices have different neighborhood color sets. The set chromatic number χ s (G) of a graph G is the minimum number of colors required in a set coloring of G. In this work, we investigate a total analog of set colorings; that is, we study set colorings of the total graph of graphs. Given a graph G = (V, E), its total graph T (G) is the graph whose vertex set is V ∪ E and in which two vertices are adjacent if and only if their corresponding elements in G are adjacent or incident. First, we establish sharp bounds for the set chromatic number of the total graph of a graph. Furthermore, we study the set colorings of the total graph of different families of graphs. Proposition 1.2 (Chartrand et al.,). For n ≥ 2 and 0 ≤ t ≤ n, χ s (G n,t ) = n.

Discrete Mathematics, 2003

We consider the subchromatic number χS(G) of graph G, which is the minimum order of all partitions of V(G) with the property that each class in the partition induces a disjoint union of cliques. Here we establish several bounds on subchromatic number. For example, we consider the maximum subchromatic number of all graphs of order n and in so doing answer a question posed in Jensen and Toft (Graph Coloring Problems, Wiley, New York, 1995). We also consider bounds on χS(G) when the size and genus of G are known. We also consider the parameter when applied to planar and outerplanar graphs. It is known that the problem of determining whether χS(G)⩽k is NP-complete for all k⩾2. We extend this by showing it is NP-complete for k=2 even when restricted to the class of planar triangle-free graphs with maximum degree four. As a corollary, we see that showing a planar triangle-free graph of maximum degree four has a 1-defective chromatic number of two is NP-complete, answering a question of Cowen et al. (J. Graph Theory 24(3) (1997) 205–219). We show that determining whether χS(G)⩽3 is NP-complete for planar graphs. We consider the subchromatic number of cartesian products of complete graphs and show a correspondence with a natural covering of matrices. We close by producing bounds on the subchromatic number in terms of chromatic number as well as the product of clique number with chromatic number. Sharpness for graphs with fixed clique size is discussed.

Annals of Pure and Applied Logic, 1998

A recursive graph is a graph whose vertex and edges sets are recursive. A highly recursive graph is a recursive graph that also has the following property: one can recursively determine the neighbors of a vertex. Both of these have been studied in the literature. We consider an intermediary notion: Let A be a set. An Arecursive graph is a recursive graph that also has the following property: one can recursively-in-A determine the neighbors of a vertex. We show that, if A is r.e. and not recursive, then there exists A-recursive graphs that are 2-colorable but not recursively k-colorable for any k. This is false for highly-recursive graphs but true for recursive graphs. Hence A-recursive graphs are closer in spirit to recursive graphs then to highly recursive graphs.

In their celebrated paper [Ramsey-Type Theorems, Discrete Appl. Math. 25 (1989) 37-52], Erd\H{o}s and Hajnal asked the following: is it true, that for any finite graph H there exists a constant c(H) such that for any finite graph G, if G does not contain complete or empty induced subgraphs of size at least |V(G)|^c(H), then H can be isomorphically embedded into G ? The positive answer has become known as the Erd\H{o}s-Hajnal conjecture. In Theorem 3.13 of the present paper we settle this conjecture in the affirmative. To do so, we are studying here the fine structure of ultraproducts of finite sets, so our investigations have a model theoretic character.

Combinatorica

A simple graph-product type construction shows that for all natural numbers r ≥ q, there exists an edge-coloring of the complete graph on 2 r vertices using r colors where the graph consisting of the union of arbitrary q color classes has chromatic number 2 q. We show that for each fixed natural number q, if there exists an edge-coloring of the complete graph on n vertices using r colors where the graph consisting of the union of arbitrary q color classes has chromatic number at most 2 q − 1, then n must be sub-exponential in r. This answers a question of Conlon, Fox, Lee, and Sudakov.

Discrete Mathematics, 1989

Tke s&chromatic number X,(G) of a graph G = (V, E) is the smallest order k of a partition {v,, v,, -* * 9 V,} of the vertices V(G) such that the subgraph (V;:) induced by each subset V;: consists of a disjoint union of complete subgraphs. By definition, Xs(G) <X(G), the chromatic number of G. This paper develops properties of this lower bound for the chromatic number.

Acta Mathematica Hungarica, 2020

An infinite graph is highly connected if the complement of any subgraph of smaller size is connected. We consider weaker versions of Ramsey's Theorem asserting that in any coloring of the edges of a complete graph there exist large highly connected subgraphs all of whose edges are colored by the same color.

Discussiones Mathematicae Graph Theory, 2007

Let G be a graph of order n with clique number ω(G), chromatic number χ(G) and independence number α(G). We show that χ(G) ≤ n+ω+1−α 2. Moreover, χ(G) ≤ n+ω−α 2 , if either ω + α = n + 1 and G is not a split graph or α+ω = n−1 and G contains no induced K ω+3 −C 5 .

Discrete Mathematics, 2012

It is known (Bollobás [4]; Kostochka and Mazurova ) that there exist graphs of maximum degree ∆ and of arbitrarily large girth whose chromatic number is at least c∆/ log ∆. We show an analogous result for digraphs where the chromatic number of a digraph D is defined as the minimum integer k so that V (D) can be partitioned into k acyclic sets, and the girth is the length of the shortest cycle in the corresponding undirected graph. It is also shown, in the same vein as an old result of Erdős , that there are digraphs with arbitrarily large chromatic number where every large subset of vertices is 2-colorable.

Journal of Graph Theory, 2003

A subgraph of a graph G is called trivial if it is either a clique or an independent set. Let q(G) denote the maximum number of vertices in a trivial subgraph of G. Motivated by an open problem of Erdős and McKay we show that every graph G on n vertices for which q(G) ≤ C log n contains an induced subgraph with exactly y edges, for every y between 0 and n δ(C). Our methods enable us also to show that under much weaker assumption, i.e., q(G) ≤ n/14, G still must contain an induced subgraph with exactly y edges, for every y between 0 and e Ω(√ log n) .