By Lovasz' proof of the Kneser conjecture, the chromatic number of a graph G is bounded from below by the index of the Z_2-space Hom(K_2,G) plus two. We show that the cohomological index of Hom(K_2,G) is also greater than the cohomological index of the Z_2-space Hom(C_{2r+1}, G) for r>0. This gives a new and simple proof of the strong form of the graph colouring theorem by Babson and Kozlov, which had been conjectured by Lovasz, and at the same time shows that it never gives a stronger bound than can be obtained by Hom(K_2, G). The proof extends ideas introduced by Zivaljevic in a previous elegant proof of a special case. We then generalise the arguments and obtain conditions under which corresponding results hold for other graphs in place of C_{2r+1}. This enables us to find an infinite family of test graphs of chromatic number 4 among the Kneser graphs.
Our main new result is a description of the Z_2-homotopy type of the direct limit of the system of all the spaces Hom(C_{2r+1}, G) in terms of the Z_2-homotopy type of Hom(K_2, G). A corollary is that the coindex of Hom(K_2, G) does not exceed the coindex of Hom(C_{2r+1}, G) by more then one if r is chosen sufficiently large. Thus the graph colouring bound in the theorem by Babson & Kozlov is also never weaker than that from Lovasz' proof of the Kneser conjecture.
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 $K_n$.
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.
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…
Soon after his 1964 seminal paper on edge colouring, Vizing asked the following question: can an optimal edge colouring be reached from any given proper edge colouring through a series of Kempe changes? We answer this question in the affirmative for triangle-free graphs.
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.
With any (not necessarily proper) edge $k$-colouring $γ:E(G)\longrightarrow\{1,\dots,k\}$ of a graph $G$,one can associate a vertex colouring $σ\_γ$ given by $σ\_γ(v)=\sum\_{e\ni v}γ(e)$.A neighbour-sum-distinguishing edge $k$-colouring is an edge colouring whose associated vertex colouring is proper.The neighbour-sum-distinguishing index of a graph $G$ is then the smallest $k$ for which $G$ admitsa neighbour-sum-distinguishing edge $k$-colouring.These notions naturally extends to total colourings of graphs that assign colours to both vertices and edges.We study in this paper equitable neighbo…
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 …
In this short note we show that every connected $2$-edge coloured cubic graph admits an $10$-colouring. This lowers the best known upper bound for the chromatic number of connected $2$-edge coloured cubic graphs.
A strong edge-colouring of a graph is a proper edge-colouring where each colour class induces a matching. It is known that every planar graph with maximum degree $Δ$ has a strong edge-colouring with at most $4Δ+4$ colours. We show that $3Δ+1$ colours suffice if the graph has girth 6, and $4Δ$ colours suffice if $Δ\geq 7$ or the girth is at least 5. In the last part of the paper, we raise some questions related to a long-standing conjecture of Vizing on proper edge-colouring of planar graphs.
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 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 $Δ=8$.
An edge colouring of a graph $G$ is called acyclic if it is proper and every cycle contains at least three colours. We show that for every $\varepsilon>0$, there exists a $g=g(\varepsilon)$ such that if $G$ has girth at least $g$ then $G$ admits an acyclic edge colouring with at most $(1+\varepsilon)Δ$ colours.
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.
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.
An edge-colouring is {\em strong} if every colour class is an induced matching. In this work we give a formulae that determines either the optimal or the optimal plus one strong chromatic index of bipartite outerplanar graphs. Further, we give an improved upper bound for any outerplanar graph which is close to optimal. All our proofs yield efficient algorithms to construct such colourings.
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.
For every fixed integer $k \geq 1$, we prove that $k$-Edge Colouring is fixed-parameter-tractable when parameterized by the number of vertices of maximum degree.
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…
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.
A graph is $H$-free if it does not contain an induced subgraph isomorphic to $H$. For every integer $k$ and every graph $H$, we determine the computational complexity of $k$-Edge Colouring for $H$-free graphs.
A graph $G$ is \emph{chordless} if no cycle in $G$ has a chord. In the present work we investigate the chromatic index and total chromatic number of chordless graphs. We describe a known decomposition result for chordless graphs and use it to establish that every chordless graph of maximum degree $Δ\geq 3$ has chromatic index $Δ$ and total chromatic number $Δ+ 1$. The proofs are algorithmic in the sense that we actually output an optimal colouring of a graph instance in polynomial time.
For a fixed graph $H$, what is the smallest number of colours $C$ such that there is a proper edge-colouring of the complete graph $K_n$ with $C$ colours containing no two vertex-disjoint colour-isomorphic copies, or repeats, of $H$? We study this function and its generalisation to more than two copies using a variety of combinatorial, probabilistic and algebraic techniques. For example, we show that for any tree $T$ there exists a constant $c$ such that any proper edge-colouring of $K_n$ with at most $c n^2$ colours contains two repeats of $T$, while there are colourings with at most $c' n^{3…
A proper edge colouring of a graph is adjacent vertex distinguishing if no two adjacent vertices see the same set of colours. Using a clever application of the Local Lemma, Hatami (2005) proved that every graph with maximum degree $Δ$ and no isolated edge has an adjacent vertex distinguishing edge colouring with $Δ+ 300$ colours, provided $Δ$ is large enough. We show that this bound can be reduced to $Δ+ 19$. This is motivated by the conjecture of Zhang, Liu, and Wang (2002) that $Δ+ 2$ colours are enough for $Δ\geq 3$.
Following a given ordering of the edges of a graph $G$, the greedy edge colouring procedure assigns to each edge the smallest available colour. The minimum number of colours thus involved is the chromatic index $χ'(G)$, and the maximum is the so-called Grundy chromatic index. Here, we are interested in the restricted case where the ordering of the edges builds the graph in a connected fashion. Let $χ_c'(G)$ be the minimum number of colours involved following such an ordering. We show that it is NP-hard to determine whether $χ_c'(G)>χ'(G)$. We prove that $χ'(G)=χ_c'(G)$ if $G$ is bipartite, and…
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.
The colouring defect of a cubic graph, introduced by Steffen in 2015, is the minimum number of edges that are left uncovered by any set of three perfect matchings. Since a cubic graph has defect $0$ if and only if it is $3$-edge-colourable, this invariant can measure how much a cubic graph differs from a $3$-edge-colourable graph. Our aim is to examine the relationship of colouring defect to oddness, an extensively studied measure of uncolourability of cubic graphs, defined as the smallest number of odd circuits in a $2$-factor. We show that there exist cyclically $5$-edge-connected snarks (cu…
An anagram is a word of the form $WP$ where $W$ is a non-empty word and $P$ is a permutation of $W$. A vertex colouring of a graph is anagram-free if no subpath of the graph is an anagram. Anagram-free graph colouring was independently introduced by Kamčev, Łuczak and Sudakov and ourselves. In this paper we introduce the study of anagram-free colourings of graph subdivisions. We show that every graph has an anagram-free $8$-colourable subdivision. The number of division vertices per edge is exponential in the number of edges. For trees, we construct anagram-free $10$-colourable subdivisions wi…
Following the recent paper which initiated the study of colour isomorphism problems for complete graphs, we obtain upper bounds for $f_2(n,H)$ for a family of graphs $H$ obtained as the $K_0$-th rooted power of a balanced rooted tree for some sufficiently large $K_0$. The proof uses the random polynomial method of Bukh. We also obtain matching lower bounds for $1$-subdivisions of the complete bipartite graph.
We study the mixed Ramsey number maxR(n,K_m,K_r), defined as the maximum number of colours in an edge-colouring of the complete graph K_n, such that K_n has no monochromatic complete subgraph on m vertices and no rainbow complete subgraph on r vertices. Improving an upper bound of Axenovich and Iverson, we show that maxR(n,K_m,K_4) <= n^{3/2}\sqrt{2m} for all m >= 3. Further, we discuss a possible way to improve their lower bound on maxR(n,K_4,K_4) based on incidence graphs of finite projective planes.
An edge-colouring of a graph is distinguishing, if the only automorphism which preserves the colouring is the identity. It has been conjectured that all but finitely many connected, finite, regular graphs admit a distinguishing edge-colouring with two colours. We show that all such graphs except $K_2$ admit a distinguishing edge-colouring with three colours. This result also extends to infinite, locally finite graphs. Furthermore, we are able to show that there are arbitrary large infinite cardinals $κ$ such that every connected $κ$-regular graph has distinguishing edge-colouring with two colo…
A k-valuation is a special type of edge k-colouring of a medial graph. Various graph polynomials, such as the Tutte, Penrose, Bollobás-Riordan, and transition polynomials, admit combinatorial interpretations and evaluations as weighted counts of k-valuations. In this paper, we consider a multivariate generating function of k-valuations. We show that this is a polynomial in k and hence defines a graph polynomial. We then show that the resulting polynomial has several desirable properties, including a recursive deletion-contraction-type definition, and specialises to the graph polynomials mentio…
The acyclic chromatic index, denoted by $a'(G)$, of a graph $G$ is the minimum number of colours used in any proper edge colouring of $G$ such that the union of any two colour classes does not contain a cycle, that is, forms a forest. We show that $a'(G\Box H)\le a'(G) + a'(H)$ for any two graphs $G$ and $H$ such that $max\{a'(G), a'(H)\} > 1$. Here, $G \Box H$ denotes the cartesian product of $G$ and $H$. This extends a recent result of [15] where tight and constructive bounds on $a'(G)$ were obtained for a class of grid-like graphs which can be expressed as the cartesian product of a number …
It was conjectured by the third author in about 1973 that every $d$-regular planar graph (possibly with parallel edges) can be $d$-edge-coloured, provided that for every odd set $X$ of vertices, there are at least $d$ edges between $X$ and its complement. For $d = 3$ this is the four-colour theorem, and the conjecture has been proved for all $d\le 7$, by various authors. Here we prove it for $d = 8$.
For a given snark G and edge e of G, we can form a cubic graph G_e using an operation we call "edge subtraction". The number of 3-edge-colourings of G_e is 18 * ψ(G,e) for some nonnegative integer ψ(G,e). Given snarks G_1 and G_2, we can form a new snark G using techniques given by Isaacs and Kochol. In this note we give relationships between ψ(G_1,e_1), ψ(G_2,e_2), and ψ(G,e) for particular edges e_1, e_2, and e, in G_1, G_2, and G (respectively). As a consequence, if g,h,i,j,k,l are each a nonnegative integer, then there exists a cyclically 5-edge-connected snark G with an edge e such that ψ…
Let $G$ be a group of orthogonal transformations of a real Hilbert space $H$. Let $R$ and $W$ be bounded $G$-stable subsets of $H$. Let $\|.\|_R$ be the seminorm on $H$ defined by $\|x\|_R:=\sup_{r\in R}|\langle r,x\rangle|$ for $x\in H$. We show that if $W$ is weakly compact and the orbit space $R^k/G$ is compact for each $k\in\oN$, then the orbit space $W/G$ is compact when $W$ is equiped with the norm topology induced by $\|.\|_R$. As a consequence we derive the existence of limits of edge-colouring models which answers a question posed by Lovász. It forms the edge-colouring counterpart of …
The Thue colouring of a graph is a colouring such that the sequence of vertex colours of any path of even and finite length in $G$ is non-repetitive. The change in the Thue number, $π(G)$, as edges are iteratively removed from a graph $G$ is studied. The notion of the $τ$-index denoted, $τ(G)$, of a graph $G$ is introduced as well. $τ(G)$ serves as a measure for the efficiency of edge deletion to reduce the Thue chromatic number of a graph.
The arboricity of a graph G is the minimum number of colours needed to colour the edges of G so that every cycle gets at least two colours. Given a positive integer p, we define the generalized p-arboricity Arb_p(G) of a graph G as the minimum number of colours needed to colour the edges of a multigraph G in such a way that every cycle C gets at least min(|C|; p + 1) colours. In the particular case where G has girth at least p + 1, Arb_p(G) is the minimum size of a partition of the edge set of G such that the union of any p parts induce a forest. If we require further that the edge colouring b…
We show that for any $2$-local colouring of the edges of the balanced complete bipartite graph $K_{n,n}$, its vertices can be covered with at most~$3$ disjoint monochromatic paths. And, we can cover almost all vertices of any complete or balanced complete bipartite $r$-locally coloured graph with $O(r^2)$ disjoint monochromatic cycles.\\ We also determine the $2$-local bipartite Ramsey number of a path almost exactly: Every $2$-local colouring of the edges of $K_{n,n}$ contains a monochromatic path on $n$ vertices.
We investigate a class of 2-edge coloured bipartite graphs known as alternating signed bipartite graphs (ASBGs) that encode the information in alternating sign matrices. The central question is when a given bipartite graph admits an ASBG-colouring; a 2-edge colouring such that the resulting graph is an ASBG. We introduce the concept of a difference-1 colouring, a relaxation of the concept of an ASBG-colouring, and present a set of necessary and sufficient conditions for when a graph admits a difference-1 colouring. The relationship between distinct difference-1 colourings of a particular graph…
A proper edge $k$-colouring of a graph $G=(V,E)$ is an assignment $c:E\to \{1,2,\ldots,k\}$ of colours to the edges of the graph such that no two adjacent edges are associated with the same colour. A neighbour sum distinguishing edge $k$-colouring, or nsd $k$-colouring for short, is a proper edge $k$-colouring such that $\sum_{e\ni u}c(e)\neq \sum_{e\ni v}c(e)$ for every edge $uv$ of $G$. We denote by $χ'_{\sum}(G)$ the neighbour sum distinguishing index of $G$, which is the least integer $k$ such that an nsd $k$-colouring of $G$ exists. By definition at least maximum degree, $Δ(G)$ colours ar…
The colouring number col(G) of a graph G is the smallest integer k for which there is an ordering of the vertices of G such that when removing the vertices of G in the specified order no vertex of degree more than k-1 in the remaining graph is removed at any step. An edge e of a graph G is said to be double-col-critical if the colouring number of G-V(e) is at most the colouring number of G minus 2. A connected graph G is said to be double-col-critical if each edge of G is double-col-critical. We characterise the double-col-critical graphs with colouring number at most 5. In addition, we prove …
We consider the problem of $q$-colouring a $k$-uniform random hypergraph, where $q,k \geq 3$, and determine the rigidity threshold. For edge densities above the rigidity threshold, we show that almost all solutions have a linear number of vertices that are linearly frozen, meaning that they cannot be recoloured by a sequence of colourings that each change the colour of a sublinear number of vertices. When the edge density is below the threshold, we prove that all but a vanishing proportion of the vertices can be recoloured by a sequence of colourings that recolour only one vertex at a time. Th…
