Related papers: Finite subgraphs of uncountably chromatic graphs
A vertex colouring of a graph is \emph{nonrepetitive} if there is no path whose first half receives the same sequence of colours as the second half. A graph is nonrepetitively $k$-choosable if given lists of at least $k$ colours at each…
Let ${\cal{F}}=\{F_1,F_2,\ldots\}$ be a sequence of graphs such that $F_n$ is a graph on $n$ vertices with maximum degree at most $\Delta$. We show that there exists an absolute constant $C$ such that the vertices of any 2-edge-colored…
Hajos' conjecture that every simple even graph on $n$ vertices can be decomposed into at most $(n-1)/2$ cycles (see L. Lovasz, On covering of graphs, in: P. Erdos, G.O.H. Katona (Eds.), Theory of Graphs, Academic Press, New York, 1968, pp.…
We improve upper bounds on the chromatic number proven independently in \cite{reedNote} and \cite{ingo}. Our main lemma gives a sufficient condition for two paths in graph to be completely joined. Using this, we prove that if a graph has an…
Archdeacon (1987) proved that graphs embeddable on a fixed surface can be $3$-coloured so that each colour class induces a subgraph of bounded maximum degree. Edwards, Kang, Kim, Oum and Seymour (2015) proved that graphs with no…
Suppose that $G$ is a graph of cardinality $\mu^+$ with chromatic number $\chi(G)\geq \mu^+$. One possible reason that this could happen is if $G$ contains a clique of size $\mu^+$. We prove that this is indeed the case when the edge…
We say that an edge colouring $c$ of a graph preserves an automorphism $\varphi$ if $\varphi$ maps each edge to an edge of the same colour. Otherwise, we say that $c$ breaks $\varphi$. We call an automorphism of a graph small if it moves…
In the paper we state and prove theorem describing the upper bound on number of the graphs that have fixed number of vertices |V| and can be colored with the fixed number of n colors. The bound relates both numbers using power of 2, while…
For an oriented graph $G$, let $f(G)$ denote the maximum chromatic number of an acyclic subgraph of $G$. Let $f(n)$ be the smallest integer such that every oriented graph $G$ with chromatic number larger than $f(n)$ has $f(G) > n$. Let…
We start by building up some theory to state Wagner's Theorem, and then prove it using Kuratowski's Theorem, a proof of which is found in Diester (2000). Following this, we establish some connections between the chromatic number of a graph…
It is shown that for any fixed $c \geq 3$ and $r$, the maximum possible chromatic number of a graph on $n$ vertices in which every subgraph of radius at most $r$ is $c$ colorable is $\tilde{\Theta}\left(n ^ {\frac{1}{r+1}} \right)$ (that…
It has been conjectured that if a finite graph has a vertex coloring such that the union of any two color classes induces a connected graph, then for every set $T$ of vertices containing exactly one member from each color class there exists…
Starting from a supercompact cardinal we build a model in which $2^{\aleph_{\omega_1}}=2^{\aleph_{\omega_1+1}}=\aleph_{\omega_1+3}$ but there is a jointly universal family of size $\aleph_{\omega_1+2}$ of graphs on $\aleph_{\omega_1+1}$.…
Given a fixed integer $n$, we prove Ramsey-type theorems for the classes of all finite ordered $n$-colorable graphs, finite $n$-colorable graphs, finite ordered $n$-chromatic graphs, and finite $n$-chromatic graphs.
A graph $G$ is called chromatic-choosable if $\chi(G)=ch(G)$. A natural problem is to determine the minimum number of vertices in a $k$-chromatic non-$k$-choosable graph. It was conjectured by Ohba, and proved by Noel, Reed and Wu that…
Let $\mathcal{C}_4(n)$ be the family of all connected $4$-chromatic graphs of order $n$. Given an integer $x\geq 4$, we consider the problem of finding the maximum number of $x$-colorings of a graph in $\mathcal{C}_4(n)$. It was conjectured…
The Hall ratio of a graph G is the maximum of |V(H)|/alpha(H) over all subgraphs H of G. Clearly, the Hall ratio of a graph is a lower bound for the fractional chromatic number. It has been asked whether conversely, the fractional chromatic…
This paper serves as the first extension of the topic of dominator colorings of graphs to the setting of digraphs. We establish the dominator chromatic number over all possible orientations of paths and cycles. In this endeavor we discover…
We prove that the statement "for every infinite cardinal nu, every graph with list chromatic nu has coloring number at most beth_omega (nu)" proved by Kojman [6] using the RGCH theorem [11] implies the RGCG theorem via a short forcing…
We develop an improved bound for the chromatic number of graphs of maximum degree $\Delta$ under the assumption that the number of edges spanning any neighbourhood is at most $(1-\sigma)\binom{\Delta}{2}$ for some fixed $0<\sigma<1$. The…