C. D. Godsil, J. Zaks
Let $G$ be the graph with the points of the unit sphere in $\mathbb{R}^3$ as its vertices, by defining two unit vectors to be adjacent if they are orthogonal as vectors. We present a proof, based on work of Hales and Straus chromatic number of this graph is four. We also prove that the subgraph of G induced by the unit vectors with rational coordinates is 3-colourable.
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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 …
Two colourings of a graph are orthogonal if they have the property that when two vertices are coloured with the same colour in one colouring, then those vertices receive distinct colours in the other colouring. In this paper, orthogonal colourings of Cayley graphs are discussed. Firstly, the orthogonal chromatic number of cycle graphs are completely determined. Secondly, the orthogonal chromatic number of certain circulant graphs is explored. Lastly, orthogonal colourings of product graphs and Hamming graphs are studied.
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.
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 $$n_0 \leq |V(G) | \le (2-ε)χ(G).$$
In this paper, we study orthogonal colourings of random geometric graphs. Two colourings of a graph are orthogonal if they have the property that when two vertices receive the same colour in one colouring, then those vertices receive distinct colours in the other colouring. A random geometric graph $RG(n,r)$ is a graph constructed by randomly placing $n$ vertices in the unit square and connecting two vertices with an edge if and only if their distance is less than the threshold $r$. We show first that random geometric graphs with $r>n^{-α}$, where $0\leq α\leq\frac{1}{4}$, have an orthogonal c…
One of the most famous applications of Graph Theory is in the field of Channel Assignment Problems. There are varieties of graph colouring concepts that are used for different requirements of frequency assignments in communication channels. We introduce here L(t, 1)-colouring of graphs. This has its foundation in T-colouring and L(p, q)-colouring. For a given finite set T including zero, an L(t, 1)-colouring of a graph G is an assignment of non-negative integers to the vertices of G such that the difference between the colours of adjacent vertices must not belong to the set T and the colours o…
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…
A homomorphism from a graph G to a graph H is a vertex mapping f from the vertex set of G to the vertex set of H such that there is an edge between vertices f(u) and f(v) of H whenever there is an edge between vertices u and v of G. The H-Colouring problem is to decide whether or not a graph G allows a homomorphism to a fixed graph H. We continue a study on a variant of this problem, namely the Surjective H-Colouring problem, which imposes the homomorphism to be vertex-surjective. We build upon previous results and show that this problem is NP-complete for every connected graph H that has exac…
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…
Let $G$ be a graph of maximum degree $Δ$ and $k$ be an integer. The $k$-recolouring graph of $G$ is the graph whose vertices are $k$-colourings of $G$ and where two $k$-colourings are adjacent if they differ at exactly one vertex. It is well-known that the $k$-recolouring graph is connected for $k\geq Δ+2$. Feghali, Johnson and Paulusma [Journal of Graph Theory, 83(4):340--358] showed that the $(Δ+1)$-recolouring graph is composed by a unique connected component of size at least $2$ and (possibly many) isolated vertices. In this paper, we study the proportion of isolated vertices (also called …
For graphs $G$ and $H$, an {\em $H$-colouring} of $G$ (or {\em homomorphism} from $G$ to $H$) is a function from the vertices of $G$ to the vertices of $H$ that preserves adjacency. $H$-colourings generalize such graph theory notions as proper colourings and independent sets. For a given $H$, $k \in V(H)$ and $G$ we consider the proportion of vertices of $G$ that get mapped to $k$ in a uniformly chosen $H$-colouring of $G$. Our main result concerns this quantity when $G$ is regular and bipartite. We find numbers $0 \leq a^-(k) \leq a^+(k) \leq 1$ with the property that for all such $G$, with h…
On Generalized Total Colourings of Planar Graphs
本文研究图的广义全染色，证实了Borowiecki等人提出的平面图广义全染色仅需4种颜色的猜想，针对两类无限平面图族完成了验证。
A vertex colouring of a graph is called asymmetric if the only automorphism which preserves it is the identity. Tucker conjectured that if every automorphism of a connected, locally finite graph moves infinitely many vertices, then there is an asymmetric colouring with $2$ colours. We make progress on this conjecture in the special case of graphs with bounded maximal degree. More precisely, we prove that if every automorphism of a connected graph with maximal degree $Δ$ moves infinitely many vertices, then there is an asymmetric colouring using $\mathcal O(\sqrt Δ\log Δ)$ colours. This is the …
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…
A clique colouring of a graph is a colouring of the vertices so that no maximal clique is monochromatic (ignoring isolated vertices). The smallest number of colours in such a colouring is the clique chromatic number. In this paper, we study the asymptotic behaviour of the clique chromatic number of the random graph G(n,p) for a wide range of edge-probabilities p=p(n). We see that the typical clique chromatic number, as a function of the average degree, forms an intriguing step function.
We consider the following two-player game, parametrised by positive integers $n$ and $k$. The game is played between Painter and Builder, alternately taking turns, with Painter moving first. The game starts with the empty graph on $n$ vertices. In each round Painter colours a vertex of her choice by one of the $k$ colours and Builder claims an edge between two previously unconnected vertices. Both players should maintain that during the game the graph admits a proper $k$-colouring. The game ends if either all $n$ vertices have been coloured, or Painter has no legal move. In the former case, Pa…
For graphs $G$ and $H$, an $H$-colouring of $G$ is a map $ψ:V(G)\rightarrow V(H)$ such that $ij\in E(G)\Rightarrowψ(i)ψ(j)\in E(H)$. The number of $H$-colourings of $G$ is denoted by $\hom(G,H)$. We prove the following: for all graphs $H$ and $δ\geq3$, there is a constant $κ(δ,H)$ such that, if $n\geqκ(δ,H)$, the graph $K_{δ,n-δ}$ maximises the number of $H$-colourings among all connected graphs with $n$ vertices and minimum degree $δ$. This answers a question of Engbers. We also disprove a conjecture of Engbers on the graph $G$ that maximises the number of $H$-colourings when the assumption o…
Recently Lau-Jeyaseeli-Shiu-Arumugam introduced the concept of the "Sudoku colourings" of graphs -- partial $χ(G)$-colourings of $G$ that have a unique extension to a proper $χ(G)$-colouring of all the vertices. They introduced the Sudoku number of a graph as the minimal number of coloured vertices in a Sudoku colouring. They conjectured that a connected graph has Sudoku number $n-1$ if, and only if, it is complete. In this note we prove that this is true.
We show that, given an infinite cardinal $μ$, a graph has colouring number at most $μ$ if and only if it contains neither of two types of subgraph. We also show that every graph with infinite colouring number has a well-ordering of its vertices that simultaneously witnesses its colouring number and its cardinality.
For any two non-negative integers h and k, h > k, an L(h, k)-colouring of a graph G is a colouring of vertices such that adjacent vertices admit colours that at least differ by h and vertices that are two distances apart admit colours that at least differ by k. The smallest positive integer δ such that G permits an L(h, k)-colouring with maximum colour δ is known as the L(h, k)-chromatic number (L(h, k)-colouring number) denoted by λ_{h,k}(G). In this paper, we discuss some interesting invariants in hypergraphs. In fact, we study the relation between the spectral gap and L(2, 1)-chromatic numb…
An ordering of the vertices of a graph is \emph{connected} if every vertex (but the first) has a neighbor among its predecessors. The greedy colouring algorithm of a graph with a connected order consists in taking the vertices in order, and assigning to each vertex the smallest available colour. A graph is \emph{good} if the greedy algorithm on every connected order gives every connected induced subgraph of it an optimal colouring. We give the characterization of good claw-free graphs in terms of minimal forbidden induced subgraphs.
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 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 look at colourings of $r$-uniform hypergraphs, focusing our attention on unique colourability and gaps in the chromatic spectrum. The pattern of an edge $E$ in an $r$-uniform hypergraph $H$ whose vertices are coloured is the partition of $r$ induced by the colour classes of the vertices in $E$. Let $Q$ be a set of partitions of $r$. A $Q$-colouring of $H$ is a colouring of its vertices such that only patterns appearing in $Q$ are allowed. We first show that many known hypergraph colouring problems, including Ramsey theory, can be stated in the language of $Q$-colourings. Then, using as our …
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…
The $k$-Strong Conflict-Free ($k$-SCF, in short) colouring problem seeks to find a colouring of the vertices of a hypergraph $H$ using minimum number of colours so that in every hyperedge $e$ of $H$, there are at least $\min\{|e|,k\}$ vertices whose colour is different from that of all other vertices in $e$. In the case of interval hypergraphs, we present an exact $P$-time algorithm for the $k$-SCF problem thus solving an open problem posed by Cheilaris et al. (2014). We achieve our results by showing that for any hypergraph a $k$-SCF colouring is a proper colouring of a related simple graph w…
Consider the following two ways to colour the vertices of a graph where the requirement that adjacent vertices get distinct colours is relaxed. A colouring has "defect" $d$ if each monochromatic component has maximum degree at most $d$. A colouring has "clustering" $c$ if each monochromatic component has at most $c$ vertices. This paper surveys research on these types of colourings, where the first priority is to minimise the number of colours, with small defect or small clustering as a secondary goal. List colouring variants are also considered. The following graph classes are studied: outerp…
The Partial List Colouring Conjecture is False
本文构造了一个14顶点、列表色数为3的图，其存在2-列表分配使最多9个顶点可正确着色，从而否证了部分列表染色猜想，反例由ChatGPT 6 Astra Ultra发现并验证。
A well-studied concept is that of the total chromatic number. A proper total colouring of a graph is a colouring of both vertices and edges so that every pair of adjacent vertices receive different colours, every pair of adjacent edges receive different colours and every vertex and incident edge receive different colours. This paper considers a strengthening of this condition and examines the minimum number of colours required for a total colouring with the additional property that for any adjacent vertices $u$ and $v$, the sets of colours incident to $u$ is different from the set of colours i…
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.
A proper vertex colouring of a graph $G$ is referred to as conflict-free if in the neighbourhood of every vertex some colour appears exactly once, while it is called $h$-conflict-free if there are at least $h$ such colours for each vertex of $G$. The least numbers of colours in such colourings of $G$ are denoted $χ_{\rm pcf}(G)$ and $χ_{\rm pcf}^h(G)$, respectively. It is known that $χ_{\rm pcf}^h(G)$ can be as large as $(h+1)(Δ+1)\approx Δ^2$ for graphs with maximum degree $Δ$ and $h$ very close to $Δ$. We provide several new upper bounds for these parameters for graphs with minimum degrees $…
One of the most interesting new developments in hypergraph colourings in these last few years has been Voloshin's notion of colourings of mixed hypergraphs. In this paper we shall study a specific instance of Voloshin's idea: a non-monochromatic non-rainbow (NMNR) colouring of a hypergraph is a colouring of its vertices such that every edge has at least two vertices coloured with different colours (non-monochromatic) and no edge has all of its vertices coloured with distinct colours (non-rainbow). Perhaps the most intriguing phenomenon of such colourings is that a hypergraph can have gaps in i…
A proper vertex colouring of a graph is \emph{nested} if the vertices of each of its colour classes can be ordered by inclusion of their open neighbourhoods. Through a relation to partially ordered sets, we show that the nested chromatic number can be computed in polynomial time. Clearly, the nested chromatic number is an upper bound for the chromatic number of a graph. We develop multiple distinct bounds on the nested chromatic number using common properties of graphs. We also determine the behaviour of the nested chromatic number under several graph operations, including the direct, Cartesia…
Asymptotically Optimal Proper Conflict-Free Colouring
本文研究了图的正常无冲突着色问题，证明了 Caro, Petruševski 和 Škrekovski 提出的关于最大度为 $Δ$ 的连通图至多使用 $Δ+1$ 种颜色的猜想渐近成立，即存在使用 $(1+o(1))Δ$ 种颜色的正常无冲突着色。
Alon et al. introduced the concept of non-repetitive colourings of graphs. Here we address some questions regarding non-repetitive colourings of planar graphs. Specifically, we show that the faces of any outerplanar map can be non-repetitively coloured using at most five colours. We also give some lower bounds for the number of colours required to non-repetitively colour the vertices of both outerplanar and planar graphs.
Maximal colourings for graphs
本文研究两种图染色概念（顶点$t$-周期染色和有向边周期染色），引入对应的染色数并分别探讨，最终揭示二者间的深刻联系。
In this paper we introduce the notion of $Σ$-colouring of a graph $G$: For given subsets $Σ(v)$ of neighbours of $v$, for every $v\in V(G)$, this is a proper colouring of the vertices of $G$ such that, in addition, vertices that appear together in some $Σ(v)$ receive different colours. This concept generalises the notion of colouring the square of graphs and of cyclic colouring of graphs embedded in a surface. We prove a general result for graphs embeddable in a fixed surface, which implies asymptotic versions of Wegner's and Borodin's Conjecture on the planar version of these two colourings. …
Consider a simple graph $G=(V,E)$ of maximum degree $Δ$ and its proper total colouring $c$ with the elements of the set $\{1,2,\ldots,k\}$. The colouring $c$ is said to be \emph{neighbour sum distinguishing} if for every pair of adjacent vertices $u$, $v$, we have $c(u)+\sum_{e\ni u}c(e)\neq c(v)+\sum_{e\ni v}c(e)$. The least integer $k$ for which it exists is denoted by $χ"_{\sum}(G)$, hence $χ"_{\sum}(G) \geq Δ+1$. On the other hand, it has been daringly conjectured that just one more label than presumed in the famous Total Colouring Conjecture suffices to construct such total colouring $c$,…
In this paper, two open conjectures are disproved. One conjecture regards independent coverings of sparse partite graphs, whereas the other conjecture regards orthogonal colourings of tree graphs. A relation between independent coverings and orthogonal colourings is established. This relation is applied to find independent coverings of some sparse partite graphs. Additionally, a degree condition providing the existence of an independent covering in the case where the graph has a square number of vertices is found.
Equitable tree colouring of graphs
研究图的树着色问题，证明当顶点数足够大时，存在公平的树着色方案，验证了相关猜想。
Let $k \geq 1$ be an integer. The reconfiguration graph $R_k(G)$ of the $k$-colourings of a graph~$G$ has as vertex set the set of all possible $k$-colourings of $G$ and two colourings are adjacent if they differ on exactly one vertex. A conjecture of Cereceda from 2007 asserts that for every integer $\ell \geq k + 2$ and $k$-degenerate graph $G$ on $n$ vertices, $R_{\ell}(G)$ has diameter $O(n^2)$. The conjecture has been verified only when $\ell \geq 2k + 1$. We give a simple proof that if $G$ is a planar graph on $n$ vertices, then $R_{10}(G)$ has diameter at most $n^2$. Since planar graphs…
A proper total $k$-colouring of a graph $G=(V,E)$ is an assignment $c : V \cup E\to \{1,2,\ldots,k\}$ of colours to the edges and the vertices of $G$ such that no two adjacent edges or vertices and no edge and its end-vertices are associated with the same colour. A total neighbour sum distinguishing $k$-colouring, or tnsd $k$-colouring for short, is a proper total $k$-colouring such that $\sum_{e\ni u}c(e)+c(u)\neq \sum_{e\ni v}c(e)+c(v)$ for every edge $uv$ of $G$. We denote by $χ''_Σ(G)$ the total neighbour sum distinguishing index of $G$, which is the least integer $k$ such that a tnsd edge…
We present results referring to the Hadwiger-Nelson problem which asks for the minimum number of colours needed to colour the plane with no two points at distance $1$ having the same colour. Exoo considered a more general problem concerning graphs $G_{[a,b]}$ with $\mathbb{R}^2$ as the vertex set and two vertices adjacent if their distance is in the interval $[a,b]$. Exoo conjectured $χ(G_{[a,b]}) = 7$ for sufficiently small but positive difference between $a$ and $b$. We partially answer this conjecture by proving that $χ(G_{[a,b]}) \geq 5$ for $b > a$. A $j$-fold colouring of graph $G = (V,E…
A lambda colouring (or $L(2,1)-$colouring) of a graph is an assignment of non-negative integers (with minimum assignment $0$) to its vertices such that the adjacent vertices must receive integers at least two apart and vertices at distance two must receive distinct integers. The lambda chromatic number (or the $λ$ number) of a graph $G$ is the least positive integer among all the maximum assigned positive integer over all possible lambda colouring of the graph $G$. Here we have primarily shown that every graph with lambda chromatic number $t$ can be embedded in a graph, with lambda chromatic n…
Linearity and Local Correctness in Weighted Colourings of Random Graphs
本文研究二项随机图的加权色数推广，证明其随最大顶点度数的线性增长依赖于边权重矩的有界性，并得出平衡子图局部正确加权染色的尖锐阈值结果。
For a positive integer $k$ and graph $G=(V,E)$, a $k$-colouring of $G$ is a mapping $c: V\rightarrow\{1,2,\ldots,k\}$ such that $c(u)\neq c(v)$ whenever $uv\in E$. The $k$-Colouring problem is to decide, for a given $G$, whether a $k$-colouring of $G$ exists. The $k$-Precolouring Extension problem is to decide, for a given $G=(V,E)$, whether a colouring of a subset of $V$ can be extended to a $k$-colouring of $G$. A $k$-list assignment of a graph is an allocation of a list -a subset of $\{1,\ldots,k\}$- to each vertex, and the List $k$-Colouring problem is to decide, for a given $G$, whether $…
We prove that the problem of counting the number of colourings of the vertices of a graph with at most two colours, such that the colour classes induce connected subgraphs is #P-complete. We also show that the closely related problem of counting the number of cocircuits of a graph is #P-complete.
Total b-chromatic Colouring of Graphs
本文引入图的全b-染色概念（扩展b-染色至顶点和边），证明一般图中计算全b-色数是NP难的，且给出毛虫图子类的多项式时间计算算法及最大颜色数的全b-染色方法。
A note on monotonicity in Maker-Breaker graph colouring games
本文研究Maker-Breaker图着色博弈的单调性，肯定回答了树状度博弈的单调性问题，证明有序顶点着色博弈不具单调性，并为连通版本的两个开放问题提供反例。
A clique colouring of a graph is a colouring of the vertices such that no maximal clique is monochromatic (ignoring isolated vertices). The least number of colours in such a colouring is the clique chromatic number. Given $n$ points $x_1, \ldots,x_n$ in the plane, and a threshold $r>0$, the corresponding geometric graph has vertex set $\{v_1,\ldots,v_n\}$, and distinct $v_i$ and $v_j$ are adjacent when the Euclidean distance between $x_i$ and $x_j$ is at most $r$. We investigate the clique chromatic number of such graphs. We first show that the clique chromatic number is at most 9 for any geom…
In 1971, Tomescu conjectured [Le nombre des graphes connexes $k$-chromatiques minimaux aux sommets étiquetés, C. R. Acad. Sci. Paris 273 (1971), 1124--1126] that every connected graph $G$ on $n$ vertices with $χ(G) = k \geq 4$ has at most $k!(k-1)^{n-k}$ $k$-colourings, where equality holds if and only if the graph is formed from $K_k$ by repeatedly adding leaves. In this note we prove (a strengthening of) the conjecture of Tomescu when $k=5$.
Temporal Role Colouring
该研究将图的角色着色扩展到时序图，证明时序角色着色问题是NP完全的，并通过参数化分析给出了该问题的固定参数可处理性结果。
Several possible definitions of local injectivity for a homomorphism of an oriented graph $G$ to an oriented graph $H$ are considered. In each case, we determine the complexity of deciding whether there exists such a homomorphism when $G$ is given and $H$ is a fixed tournament on three or fewer vertices. Each possible definition leads to a locally-injective oriented colouring problem. A dichotomy theorem is proved in each case.
We show that if $\gS=(V,E)$ is a regular bipartite graph for which the expansion of subsets of a single parity of $V$ is reasonably good and which satisfies a certain local condition (that the union of the neighbourhoods of adjacent vertices does not contain too many pairwise non-adjacent vertices), and if $\cM$ is a Markov chain on the set of proper 3-colourings of $\gS$ which updates the colour of at most $ρ|V|$ vertices at each step and whose stationary distribution is uniform, then for $ρ\approx .22$ and $d$ sufficiently large the convergence to stationarity of $\cM$ is (essentially) expon…
Reconstructing a graph from its Bell colouring graph
本文研究了图的贝尔着色图的同构条件，证明了特定条件下图可通过贝尔着色图唯一确定。
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 …
We give a short proof of the following theorem due to Jon H. Folkman (1969): The chromatic number of any graph is at most $2$ plus the maximum over all subgraphs of the difference between half the number of vertices and the independence number.
A distinguishing colouring of a graph is a colouring of the vertex set such that no non-trivial automorphism preserves the colouring. Tucker conjectured that if every non-trivial automorphism of a locally finite graph moves infinitely many vertices, then there is a distinguishing 2-colouring. We show that the requirement of local finiteness is necessary by giving a non-locally finite graph for which no finite number of colours suffices.
For a fixed integer, the $k$-Colouring problem is to decide if the vertices of a graph can be coloured with at most $k$ colours for an integer $k$, such that no two adjacent vertices are coloured alike. A graph $G$ is $H$-free if $G$ does not contain $H$ as an induced subgraph. It is known that for all $k\geq 3$, the $k$-Colouring problem is NP-complete for $H$-free graphs if $H$ contains an induced claw or cycle. The case where $H$ contains a cycle follows from the known result that the problem is NP-complete even for graphs of arbitrarily large fixed girth. We examine to what extent the situ…
Stable cuts, NAC-colourings and flexible realisations of graphs
本文证明了极小刚性图存在正边长柔性实现当且仅当它不是2-树，证实了相关猜想，并基于无稳定割图刻画加强了对柔性图必有稳定割的结论，同时给出了任意图NAC-染色数量的上界及呈指数级增长的图族构造。
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