Related papers: The reverse mathematics of Brooks' theorem
We prove that there is a Borel quasi-kernel in any locally countable Borel directed graph with finite Borel chromatic number. We prove that the Borel chromatic number of a Borel directed graph with bounded out-degree $n$ is either infinite…
In this paper we study Cartesian products of graphs and their divisorial gonality, which is a tropical version of the gonality of an algebraic curve. We present an upper bound on the gonality of the Cartesian product of any two graphs, and…
In this note we give two proofs of Brooks' Theorem. The first is obtained by modifying an earlier proof and the second by combining two earlier proofs. We believe these proofs are easier to teach in Computer Science courses.
The acyclic chromatic number of a graph is the least number of colors needed to properly color its vertices so that none of its cycles has only two colors. We show that for all $\alpha>2^{-1/3}$ there exists an integer $\Delta_{\alpha}$…
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…
The Burning Number Conjecture claims that for every connected graph $G$ of order $n,$ its burning number satisfies $b(G) \le \lceil \sqrt{n} \rceil.$ While the conjecture remains open, we prove that it is asymptotically true when the order…
Motivated by recent work on majority edge-colourings of graphs, we initiate the study of the corresponding problem for hypergraphs. First, sharpening the probabilistic argument by a $KL$ large-deviation estimate, we obtain a sufficient…
The theory of convergent graph sequences has been worked out in two extreme cases, dense graphs and bounded degree graphs. One can define convergence in terms of counting homomorphisms from fixed graphs into members of the sequence…
In the List $k$-Coloring problem we are given a graph whose every vertex is equipped with a list, which is a subset of $\{1,\ldots,k\}$. We need to decide if $G$ admits a proper coloring, where every vertex receives a color from its list.…
We define an on-line (incremental) algorithm that, given a (possibly infinite) pseudo-transitive oriented graph, produces a transitive reorientation. This implies that a theorem of Ghouila-Houri is provable in RCA_0 and hence is computably…
For a graph $G$, let $\cn(G)$ and $\la(G)$ denote the chromatic number of $G$ and the maximum local edge connectivity of $G$, respectively. A result of Dirac \cite{Dirac53} implies that every graph $G$ satisfies $\cn(G)\leq \la(G)+1$. In…
We consider the problem of coloring the squares of graphs of bounded maximum average degree, that is, the problem of coloring the vertices while ensuring that two vertices that are adjacent or have a common neighbour receive different…
In a 1977 paper, Steffens identified an elegant criterion for determining when a countable graph has a perfect matching. In this paper, we will investigate the proof-theoretic strength of this result and related theorems. We show that a…
The $r$-colour Ramsey number $R_r(k)$ is the minimum $n \in \mathbb{N}$ such that every $r$-colouring of the edges of the complete graph $K_n$ on $n$ vertices contains a monochromatic copy of $K_k$. We prove, for each fixed $r \geqslant 2$,…
The generalized coloring numbers of Kierstead and Yang (Order 2003) offer an algorithmically-useful characterization of graph classes with bounded expansion. In this work, we consider the hardness and approximability of these parameters.…
We prove a version of the Loebl-Komlos-Sos Conjecture for dense graphs. For each q>0 there exists a number $n_0\in \mathbb{N}$ such that for any n>n_0 and k>qn the following holds: if G be a graph of order n with at least n/2 vertices of…
A digraph $D$ is an oriented graph if $D$ does not have a pair of opposite arcs. The degree of a vertex $v$ of $D$ is the sum of the in-degree and out-degree of $v.$ Let $fvs(D)$ be the minimum number of vertices whose deletion from $D$…
We attempt to generalize a theorem of Nash-Williams stating that a graph has a $k$-arc-connected orientation if and only if it is $2k$-edge-connected. In a strongly connected digraph we call an arc {\it deletable} if its deletion leaves a…
This paper uses the framework of reverse mathematics to investigate the strength of two recurrence theorems of topological dynamics. It establishes that one of these theorems, the existence of an almost periodic point, lies strictly between…
A subgraph of an edge-coloured graph is called rainbow if all its edges have different colours. We prove a rainbow version of the blow-up lemma of Koml\'os, S\'ark\"ozy and Szemer\'edi that applies to almost optimally bounded colourings. A…