The famous List Colouring Conjecture from the 1970s states that for every graph $G$ the chromatic index of $G$ is equal to its list chromatic index. In 1996 in a seminal paper, Kahn proved that the List Colouring Conjecture holds asymptotically. Our main result is a local generalization of Kahn's theorem. More precisely, we show that, for a graph $G$ with sufficiently large maximum degree $Δ$ and minimum degree $δ\geq \ln^{25} Δ$, the following holds: for every assignment of lists of colours to the edges of $G$, such that $|L(e)| \geq (1+o(1)) \cdot \max\left\{\rm{deg}(u),\rm{deg}(v)\right\}$ for each edge $e=uv$, there is an $L$-edge-colouring of $G$. Furthermore, Kahn showed that the List Colouring Conjecture holds asymptotically for linear, $k$-uniform hypergraphs, and recently Molloy generalized Kahn's original result to correspondence colouring as well as its hypergraph generalization. We prove local versions of all of these generalizations by showing a weighted version that simultaneously implies all of our results.
We prove that the list chromatic index of a graph of maximum degree $Δ$ and treewidth $\leq \sqrt{2Δ} -3$ is $Δ$; and that the total chromatic number of a graph of maximum degree $Δ$ and treewidth $\leq Δ/3 +1$ is $Δ+1$. This improves results by Meeks and Scott.
This paper proves that for each positive integer $m$, there is a planar graph $G$ which is not $(4m+\lfloor \frac{2m-1}{9}\rfloor,m)$-choosable. Then we pose some conjectures concerning multiple list colouring of planar graphs.
This paper proves that for each positive integer $m$, there is a triangle-free planar graph $G$ which is not $(3m+ \lceil \frac m{17} \rceil-1, m)$-choosable.
This paper disproves a conjecture of Wang, Wu, Yan and Xie, and answers in negative a question in Dvorak, Pekarek and Sereni. In return, we pose five open problems.
This paper proves the following result: If $G$ is a planar graph and $L$ is a $4$-list assignment of $G$ such that $|L(x) \cap L(y)| \le 2$ for every edge $xy$, then $G$ is $L$-colourable. This answers a question asked by Kratochvíl, Tuza and Voigt in [Journal of Graph Theory, 27(1):43--49, 1998].
The notion of a Galvin orientation of a line graph is introduced, generalizing the idea used by Galvin in his landmark proof of the list-edge-colouring conjecture for bipartite graphs. If L(G) has a proper Galvin orientation with respect to k, then it immediately implies that G is k-list-edge-colourable, but the converse is not true. The stronger property is studied in graphs of the form `bipartite plus an edge', the Petersen graph, cliques, and simple graphs without odd cycles of length 5 or longer.
In 1977, Wegner conjectured that the chromatic number of the square of every planar graph $G$ with maximum degree $Δ\ge8$ is at most $\bigl\lfloor\frac32Δ\bigr\rfloor+1$. We show that it is at most $\frac32 Δ(1+o(1))$ (where the $o(1)$ is as $Δ\to+\infty$), and indeed that this is true for the list chromatic number and for more general classes of graphs.
Let $G$ be a graph on $n$ vertices and let $\mathcal{L}_k$ be an arbitrary function that assigns each vertex in $G$ a list of $k$ colours. Then $G$ is $\mathcal{L}_k$-list colourable if there exists a proper colouring of the vertices of $G$ such that every vertex is coloured with a colour from its own list. We say $G$ is $k$-choosable if for every such function $\mathcal{L}_k$, $G$ is $\mathcal{L}_k$-list colourable. The minimum $k$ such that $G$ is $k$-choosable is called the list chromatic number of $G$ and is denoted by $χ_L(G)$. Let $χ_L(G) = s$ and let $t$ be a positive integer less than …
Ohba has conjectured \cite{ohb} that if the graph $G$ has $2χ(G)+1$ or fewer vertices then the list chromatic number and chromatic number of $G$ are equal. In this paper we prove that this conjecture is asymptotically correct. More precisely we obtain that for any $0<ε<1$, there exist an $n_0=n_0(ε)$ such that the list chromatic number of $G$ equals its chromatic number, provided $$n_0 \leq |V(G) | \le (2-ε)χ(G).$$
Let $G$ be a simple planar graph of maximum degree $Δ$, let $t$ be a positive integer, and let $L$ be an edge list assignment on $G$ with $|L(e)| \geq Δ+t$ for all $e \in E(G)$. We prove that if $H$ is a subgraph of $G$ that has been $L$-edge-coloured, then the edge-precolouring can be extended to an $L$-edge-colouring of $G$, provided that $H$ has maximum degree $d\leq t$ and either $d \leq t-4$ or $Δ$ is large enough ($Δ\geq 16+d$ suffices). If $d>t$, there are examples for any choice of $Δ$ where the extension is impossible.
We generalize a framework of list colouring results to correspondence colouring. Correspondence colouring is a generalization of list colouring wherein we localize the meaning of the colours available to each vertex. As pointed out by Dvořák and Postle, both of Thomassen's theorems on the 5-choosability of planar graphs and 3-choosability of planar graphs of girth at least five carry over to the correspondence colouring setting. In this paper, we show that the family of graphs that are critical for 5-correspondence colouring as well as the family of graphs of girth at least five that are criti…
We restate theorems of Hutchinson on list-colouring extendability for outerplanar graphs in terms of non-vanishing monomials in a graph polynomial, which yields an Alon-Tarsi equivalent for her work. This allows to simplify her proofs as well as obtain more general results.
We prove two distinct and natural refinements of a recent breakthrough result of Molloy (and a follow-up work of Bernshteyn) on the (list) chromatic number of triangle-free graphs. In both our results, we permit the amount of colour made available to vertices of lower degree to be accordingly lower. One result concerns list colouring and correspondence colouring, while the other concerns fractional colouring. Our proof of the second illustrates the use of the hard-core model to prove a Johansson-type result, which may be of independent interest.
A list assignment $L$ for a graph $G$ is an $(\ell,k)$-list assignment if $|L(v)|\geq \ell$ for each $v \in V(G)$ and $|L(u) \cap L(v)| \leq k$ for each $uv \in E(G)$. We say $G$ is $(\ell,k)$-choosable if it admits an $L$-colouring for every $(\ell, k)$-list assignment $L$. We prove that if $G$ is a planar graph with $(4,2)$-list assignment $L$ and for every triangle $T \subseteq G$ we have that $|\bigcap_{v \in V(T)} L(v)| \neq 2$, then $G$ is $L$-colourable. In fact, we prove a slightly stronger result: if $G$ contains a clique $H$ such that $V(H) \cap V(T) \neq \emptyset$ for every triangl…
Following problems posed by Gyárfás, we show that for every $r$-edge-colouring of $K_n$ there is a monochromatic triple star of order at least $n/(r-1)$, improving a previous result by Ruszinkó. An edge colouring of a graph is called a local $r$-colouring if every vertex spans edges of at most $r$ distinct colours. We prove the existence of a monochromatic triple star with at least $rn/(r^2-r+1)$ vertices in every local $r$-colouring of $K_n$.
A vertex colouring of some graph is called perfect if each vertex of colour $i$ has exactly $a_{ij}$ neighbours of colour $j$. Being perfect imposes several restrictions on the colour incidence matrix $(a_{ij})$. We list several (old and new) necessary conditions for a matrix to be the colour incidence matrix of a perfect colouring. Moreover we show that a certain combination of these conditions is also sufficient. Using this we determine a list of all colour incidence matrices corresponding to perfect colourings of 3-regular, 4-regular and 5-regular graphs with two, three and four colours, re…
Let $c$ and $c'$ be edge or vertex colourings of a graph $G$. We say that $c'$ is less symmetric than $c$ if the stabiliser (in $\operatorname{Aut} G$) of $c'$ is contained in the stabiliser of $c$. We show that if $G$ is not a bicentred tree, then for every vertex colouring of $G$ there is a less symmetric edge colouring with the same number of colours. On the other hand, if $T$ is a tree, then for every edge colouring there is a less symmetric vertex colouring with the same number of edges. Our results can be used to characterise those graphs whose distinguishing index is larger than their d…
A graph $G$ is called $3$-choice critical if $G$ is not $2$-choosable but any proper subgraph is $2$-choosable. A characterization of $3$-choice critical graphs was given by Voigt in [On list Colourings and Choosability of Graphs, Habilitationsschrift, Tu Ilmenau(1998)]. Voigt conjectured that if $G$ is a bipartite $3$-choice critical graph, then $G$ is $(4m, 2m)$-choosable for every integer $m$. This conjecture was disproved by Meng, Puleo and Zhu in [On (4, 2)-Choosable Graphs, Journal of Graph Theory 85(2):412-428(2017)]. They showed that if $G=Θ_{r,s,t}$ where $r,s,t$ have the same parity …
The Unfriendly Partition Conjecture posits that every countable graph admits a 2-colouring in which for each vertex there are at least as many bichromatic edges containing that vertex as monochromatic ones. This is not known in general, but it is known that a 3-colouring with this property always exists. Anholcer, Bosek and Grytczuk recently gave a list-colouring version of this conjecture, and proved that such a colouring exists for lists of size 4. We improve their result to lists of size 3; the proof extends to directed acyclic graphs. We also discuss some generalisations.
Let ${\mathcal D}_d$ be the class of $d$-degenerate graphs and let $L$ be a list assignment for a graph $G$. A colouring of $G$ such that every vertex receives a colour from its list and the subgraph induced by vertices coloured with one color is a $d$-degenerate graph is called the $(L,{\mathcal D}_d)$-colouring of $G$. For a $k$-uniform list assignment $L$ and $d\in\mathbb{N}_0$, a graph $G$ is equitably $(L,{\mathcal D}_d)$-colorable if there is an $(L,{\mathcal D}_d)$-colouring of $G$ such that the size of any colour class does not exceed $\left\lceil|V(G)|/k\right\rceil$. An equitable $(L…
Král' and Sgall (2005) introduced a refinement of list colouring where every colour list must be subset to one predetermined palette of colours. We call this $(k,\ell)$-choosability when the palette is of size at most $\ell$ and the lists must be of size at least $k$. They showed that, for any integer $k\ge 2$, there is an integer $C=C(k,2k-1)$, satisfying $C = O(16^{k}\ln k)$ as $k\to \infty$, such that, if a graph is $(k,2k-1)$-choosable, then it is $C$-choosable, and asked if $C$ is required to be exponential in $k$. We demonstrate it must satisfy $C = Ω(4^k/\sqrt{k})$. For an integer $\ell…
It is proved that all 4-edge-colourings of a (sub)cubic graph are Kempe equivalent. This resolves a conjecture of the second author. In fact, it is found that the maximum degree Delta=3 is a threshold for Kempe equivalence of (Delta+1)-edge-colourings, as such an equivalence does not hold in general when Delta=4. One extra colour allows a similar result in this latter case however, namely, when Delta<=4 it is shown that all (Delta+2)-edge-colourings are Kempe equivalent.
For integer q>1, we derive edge q-colouring models for (i) the Tutte polynomial of a graph G on the hyperbola H_q, (ii) the symmetric weight enumerator of the set of group-valued q-flows of G, and (iii) a more general vertex colouring model partition function that includes these polynomials and the principal specialization order q of Stanley's symmetric monochrome polynomial. In the second half of the paper we exhibit a family of non-symmetric edge q-colouring models defined on k-regular graphs, whose partition functions for q >= k each evaluate the number of proper edge k-colourings of G when…
The distinguishing index $D'(G)$ of a graph $G$ is the least number of colours needed in an edge colouring which is not preserved by any non-trivial automorphism. Broere and Pilśniak conjectured that if every non-trivial automorphism of a countable graph $G$ moves infinitely many edges, then $D'(G) \leq 2$. We prove this conjecture.
The Four Colour Theorem asserts that the vertices of every plane graph can be properly coloured with four colors. Fabrici and Göring conjectured the following stronger statement to also hold: the vertices of every plane graph can be properly coloured with the numbers 1,...,4 in such a way that every face contains a unique vertex coloured with the maximal color appearing on that face. They proved that every plane graph has such a colouring with the numbers 1,...,6. We prove that every plane graph has such a colouring with the numbers 1,...,5 and we also prove the list variant of the statement f…
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 Komlós, Sárközy and Szemerédi that applies to almost optimally bounded colourings. A corollary of this is that there exists a rainbow copy of any bounded-degree spanning subgraph $H$ in a quasirandom host graph $G$, assuming that the edge-colouring of $G$ fulfills a boundedness condition that is asymptotically best possible. This has many applications beyond rainbow colourings, for example to graph decompositions, orthogonal double covers and graph …
A total colouring of a graph is a colouring of its vertices and edges such that no two adjacent vertices or edges have the same colour and moreover, no edge coloured $c$ has its endvertex coloured $c$ too. A weak total Thue colouring of a graph $G$ is a colouring of its vertices and edges such that the colour sequence of consecutive vertices and edges of every path of $G$ is nonrepetitive. In a total Thue colouring also the induced vertex-colouring and edge-colouring of $G$ are nonrepetitive. The weak total Thue number $π_{T_w}(G)$ of a graph $G$ denotes the minimum number of colours required …
Let $K_{n,n}$ be the complete bipartite graph with $n$ vertices in each side. For each vertex draw uniformly at random a list of size $k$ from a base set $S$ of size $s=s(n)$. In this paper we estimate the asymptotic probability of the existence of a proper colouring from the random lists for all fixed values of $k$ and growing $n$. We show that this property exhibits a sharp threshold for $k\geq 2$ and the location of the threshold is precisely $s(n)=2n$ for $k=2$, and approximately $s(n)=\frac{n}{2^{k-1}\ln 2}$ for $k\geq 3$.
Let $χ_l(G)$ denote the list chromatic number of the $r$-uniform hypergraph~$G$. Extending a result of Alon for graphs, Saxton and the second author used the method of containers to prove that, if $G$ is simple and $d$-regular, then $χ_l(G)\ge (1/(r-1)+o(1))\log_r d$. To see how close this inequality is to best possible, we examine $χ_l(G)$ when $G$ is a random $r$-partite hypergraph with $n$ vertices in each class. The value when $r=2$ was determined by Alon and Krivelevich, here we show that $χ_l(G)= (g(r,α)+o(1))\log_r d$ almost surely, where $d$ is the expected average degree of~$G$ and $α…
A fundamental connection between list vertex colourings of graphs and Property B (also known as hypergraph 2-colourability) was already known to Erdős, Rubin and Taylor. In this article, we draw similar connections for improper list colourings. This extends results of Kostochka, Alon, and Král' and Sgall for, respectively, multipartite graphs, graphs of large minimum degree, and list assignments with bounded list union.
There are two conjectures concerning planar graph colourings that are strengthenings of the four colour theorem. One concerns signed graph colouring and is proposed by Máčajová, Raspaud and Škoviera. It asserts that every signed planar graph is $4$-colourable. Another concerns list colouring and is proposed by Kündgen and Ramamurthi which asserts that if $L$ is a $2$-list assignment of a planar graph $G$, then there is an $L$-colouring of $G$ such that each colour class induces a bipartite graph. In this note we prove that the first conjecture implies the second one.
Let $C_n^k$ be the $k$-th power of a cycle on $n$ vertices (i.e. the vertices of $C_n^k$ are those of the $n$-cycle, and two vertices are connected by an edge if their distance along the cycle is at most $k$). For each vertex draw uniformly at random a subset of size $c$ from a base set $S$ of size $s=s(n)$. In this paper we solve the problem of determining the asymptotic probability of the existence of a proper colouring from the lists for all fixed values of $c,k$, and growing $n$.
The main technique used to edge-colour graphs requiring $Δ+2$ or more colours is the method of Tashkinov trees. We present a specific limit to this method, in terms of Kempe changes. We also provide a new Tashkinov tree extension.
The Petersen colouring conjecture states that every bridgeless cubic graph admits an edge-colouring with $5$ colours such that for every edge $e$, the set of colours assigned to the edges adjacent to $e$ has cardinality either $2$ or $4$, but not $3$. We prove that every bridgeless cubic graph $G$ admits an edge-colouring with $4$ colours such that at most $\frac45\cdot|V(G)|$ edges do not satisfy the above condition. This bound is tight and the Petersen graph is the only connected graph for which the bound cannot be decreased. We obtain such a $4$-edge-colouring by using a carefully chosen su…
An edge colouring of a graph is said to be an $r$-local colouring if the edges incident to any vertex are coloured with at most $r$ colours. Generalising a result of Bessy and Thomassé, we prove that the vertex set of any $2$-locally coloured complete graph may be partitioned into two disjoint monochromatic cycles of different colours. Moreover, for any natural number $r$, we show that the vertex set of any $r$-locally coloured complete graph may be partitioned into $O(r^2 \log r)$ disjoint monochromatic cycles. This generalises a result of Erdős, Gyárfás and Pyber.
An adjacent vertex distinguishing edge colouring of a graph $G$ without isolated edges is its proper edge colouring such that no pair of adjacent vertices meets the same set of colours in $G$. We show that such colouring can be chosen from any set of lists associated to the edges of $G$ as long as the size of every list is at least $Δ+CΔ^{\frac{1}{2}}(\logΔ)^4$, where $Δ$ is the maximum degree of $G$ and $C$ is a constant. The proof is probabilistic. The same is true in the environment of total colourings.
An edge-face colouring of a plane graph with edge set $E$ and face set $F$ is a colouring of the elements of $E \cup F$ such that adjacent or incident elements receive different colours. Borodin proved that every plane graph of maximum degree $Δ\ge10$ can be edge-face coloured with $Δ+1$ colours. Borodin's bound was recently extended to the case where $Δ=9$. In this paper, we extend it to the case $Δ=8$.
We examine the computational complexity of approximately counting the list H-colourings of a graph. We discover a natural graph-theoretic trichotomy based on the structure of the graph H. If H is an irreflexive bipartite graph or a reflexive complete graph then counting list H-colourings is trivially in polynomial time. Otherwise, if H is an irreflexive bipartite permutation graph or a reflexive proper interval graph then approximately counting list H-colourings is equivalent to #BIS, the problem of approximately counting independent sets in a bipartite graph. This is a well-studied problem wh…
We say that a vertex colouring $φ$ of a graph $G$ is nonrepetitive if there is no positive integer $n$ and a path on $2n$ vertices $v_{1}\ldots v_{2n}$ in $G$ such that the associated sequence of colours $φ(v_{1})\ldotsφ(v_{2n})$ satisfy $φ(v_{i})=φ(v_{i+n})$ for all $i=1,2,\dots,n$. The minimum number of colours in a nonrepetitive vertex colouring of $G$ is the Thue chromatic number $π(G)$. For the case of vertex list colourings the Thue choice number $π_{l}(G)$ of $G$ denotes the smallest integer $k$ such that for every list assignment $L:V(G)\rightarrow 2^{\mathbb{N}}$ with minimum list len…
Consider any dense r-regular quasirandom bipartite graph H with parts of size n and fix a set of r colours. Let L be a random list assignment where each colour is available for each edge of H with probability p. We show that the threshold probability for H to have a proper L-edge-colouring is p of order (log n)/n. This answers a question of Kang, Kelly, Kühn, Methuku and Osthus. We thus obtain the same threshold for Steiner Triple Systems and Latin squares; the latter answers a question of Johanssen from 2006.
We survey results concerning reconfigurations of colourings and dominating sets in graphs. The vertices of the $k$-colouring graph $\mathcal{C}_{k}(G)$ of a graph $G$ correspond to the proper $k$-colourings of a graph $G$, with two $k$-colourings being adjacent whenever they differ in the colour of exactly one vertex. Similarly, the vertices of the $k$-edge-colouring graph $\mathcal{EC}_{k}(G)$ of $g$ are the proper $k$-edge-colourings of $G$, where two $k$-edge-colourings are adjacent if one can be obtained from the other by switching two colours along an edge-Kempe chain, i.e., a maximal two…
An edge colouring of a multigraph can be thought of as a partition of the edges into matchings (a matching meets each vertex at most once). Analogously, an edge cover colouring is a partition of the edges into edge covers (an edge cover meets each vertex at least once). We aim to determine a tight lower bound on the maximum number of parts in an edge cover colouring as a function of the minimum degree delta, which would be an analogue of Shannon's theorem from 1949 on edge-colouring multigraphs. We are able to give a lower bound that is tight except when delta=9 or delta is odd and > 12; in th…
A colouring of a graph $G=(V,E)$ is a function $c: V\rightarrow\{1,2,\ldots \}$ such that $c(u)\neq c(v)$ for every $uv\in E$. A $k$-regular list assignment of $G$ is a function $L$ with domain $V$ such that for every $u\in V$, $L(u)$ is a subset of $\{1, 2, \dots\}$ of size $k$. A colouring $c$ of $G$ respects a $k$-regular list assignment $L$ of $G$ if $c(u)\in L(u)$ for every $u\in V$. A graph $G$ is $k$-choosable if for every $k$-regular list assignment $L$ of $G$, there exists a colouring of $G$ that respects $L$. We may also ask if for a given $k$-regular list assignment $L$ of a given g…
In this note we prove the conjecture of \cite{HaWiWi} that every bipartite multigraph with integer edge delays admits an edge colouring with $d+1$ colours in the special case where $d=3$. A connection to the Brualdi-Ryser-Stein conjecture is discussed.
In this paper, the on-line list colouring of binomial random graphs G(n,p) is studied. We show that the on-line choice number of G(n,p) is asymptotically almost surely asymptotic to the chromatic number of G(n,p), provided that the average degree d=p(n-1) tends to infinity faster than (log log n)^1/3(log n)^2n^(2/3). For sparser graphs, we are slightly less successful; we show that if d>(log n)^(2+epsilon) for some epsilon>0, then the on-line choice number is larger than the chromatic number by at most a multiplicative factor of C, where C in [2,4], depending on the range of d. Also, for d=O(1…
