Andrea Freschi, Joseph Hyde, Joanna Lada, Andrew Treglown
Balogh, Csaba, Jing and Pluhár recently determined the minimum degree threshold that ensures a $2$-coloured graph $G$ contains a Hamilton cycle of significant colour bias (i.e., a Hamilton cycle that contains significantly more than half of its edges in one colour). In this short note we extend this result, determining the corresponding threshold for $r$-colourings.
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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.
A rainbow matching in an edge-coloured graph is a matching such that its edges have distinct colours. We show that every properly edge-coloured graph $G$ with $|G| \ge (9δ(G) -5)/2$ has a rainbow matching of size $δ(G)$, improving a result of Diemunsch et al.
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.
It is shown that Halin graphs are $Δ$-edge-choosable and that graphs of tree-width 3 are $(Δ+1)$-edge-choosable and $(Δ+2)$-total-colourable.
Norine's antipodal-colouring conjecture, in a form given by Feder and Subi, asserts that whenever the edges of the discrete cube are 2-coloured there must exist a path between two opposite vertices along which there is at most one colour change. The best bound to date was that there must exist such a path with at most $n/2$ colour changes. Our aim in this note is to improve this upper bound to $(\frac{3}{8}+o(1))n$.
A rainbow subgraph in an edge-coloured graph is a subgraph such that its edges have distinct colours. The minimum colour degree of a graph is the smallest number of distinct colours on the edges incident with a vertex over all vertices. Kostochka, Pfender, and Yancey showed that every edge-coloured graph on $n$ vertices with minimum colour degree at least $k$ contains a rainbow matching of size at least $k$, provided $n\geq (17/4)k^2$. In this paper, we show that $n\geq 4k-4$ is sufficient for $k \ge 4$.
We prove that a random graph $G(n,p)$, with $p$ above the Hamiltonicity threshold, is typically such that for any $r$-colouring of its edges there exists a Hamilton cycle with at least $(2/(r+ 1)-o(1))n$ edges of the same colour. This estimate is asymptotically optimal.
A note on colour-bias perfect matchings in hypergraphs
该研究将2着色图中保证显著色偏完美匹配存在的最小度阈值结果推广至k一致超图，确定了r着色k一致超图中对应最小ℓ度阈值。
We show that there is a constant $C$ such that for every $\varepsilon>0$ any $2$-coloured $K_n$ with minimum degree at least $n/4+\varepsilon n$ in both colours contains a complete subgraph on $2t$ vertices where one colour class forms a $K_{t,t}$, provided that $n\geq \varepsilon^{-Ct}$. Also, we prove that if $K_n$ is $2$-coloured with minimum degree at least $\varepsilon n$ in both colours then it must contain one of two natural colourings of a complete graph. Both results are tight up to the value of $C$ and they answer two recent questions posed by Kamčev and Müyesser.
Let $K_n^c$ be an edge-coloured complete graph on $n$ vertices. Let $Δ_{\rm mon}(K_n^c)$ denote the largest number of edges of the same colour incident with a vertex of $K_n^c$. A properly coloured cycle is a cycle such that no two adjacent edges have the same colour. In 1976, Bollobás and Erdős conjectured that every $K_n^c$ with $Δ_{\rm mon}(K_n^c) < \lfloor n/2 \rfloor$ contains a properly coloured Hamiltonian cycle. In this paper, we show that for any $\varepsilon > 0 $, there exists an integer $n_0$ such that every $K_n^c$ with $Δ_{\rm mon}(K_n^c) < (1/2 - \varepsilon) n $ and $n \ge n_0$…
A note on graphs of $k$-colourings
本文证明每个图都可由某个$k$-着色图唯一确定，证实相关猜想，并表明有限个广义色多项式无法唯一反推原图。
A minimum-degree threshold for colour-biased Hamilton cycles in hypergraphs
该研究确定了使双色3-超图存在颜色偏置紧哈密顿环的渐近最优最小顶点度条件，证实了相关学者提出的猜想。
A vertex colouring of a graph is \emph{nonrepetitive on paths} if there is no path $v_1,v_2,...,v_{2t}$ such that v_i and v_{t+i} receive the same colour for all i=1,2,...,t. We determine the maximum density of a graph that admits a k-colouring that is nonrepetitive on paths. We prove that every graph has a subdivision that admits a 4-colouring that is nonrepetitive on paths. The best previous bound was 5. We also study colourings that are nonrepetitive on walks, and provide a conjecture that would imply that every graph with maximum degree $Δ$ has a $f(Δ)$-colouring that is nonrepetitive on w…
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.
Akbari, Etesami, Mahini, and Mahmoody conjectured that every proper edge colouring of $K_n$ with $n$ colours contains a Hamilton cycle with $\leq O(\log n)$ colours. They proved that there is always a Hamilton cycle with $\leq 8\sqrt n$ colours. In this note we improve this bound to $O(\log^3 n)$.
A subgraph $H$ of an edge-coloured graph is called rainbow if all of the edges of $H$ have different colours. In 1989, Andersen conjectured that every proper edge-colouring of $K_{n}$ admits a rainbow path of length $n-2$. We show that almost all optimal edge-colourings of $K_{n}$ admit both (i) a rainbow Hamilton path and (ii) a rainbow cycle using all of the colours. This result demonstrates that Andersen's Conjecture holds for almost all optimal edge-colourings of $K_{n}$ and answers a recent question of Ferber, Jain, and Sudakov. Our result also has applications to the existence of transve…
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…
Hamiltonicity of Bell and Stirling Colour Graphs
本文研究了贝尔和斯特林着色图的哈密顿性，证明了除特殊情况外，这些图具有哈密顿回路。
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.
Colour-biased Hamilton cycles in randomly perturbed graphs
本文研究随机扰动图中颜色偏置哈密顿圈的存在性，证明添加 $O(n)$ 条随机边可在最小度条件下保证 $Ω(n)$ 颜色偏置，并刻画了临界端点处 $Ω(m)$ 颜色偏置的相变行为。
Multi-Colouring of Kneser Graphs: Notes on Stahl's Conjecture
本文研究图的多着色参数间蕴含关系，将其转化为 Kneser 图多着色问题，围绕 Stahl 1976 年猜想给出新结果、强化已有结论并简化若干已知情形的证明。
Given an $n$-vertex graph $G$ with minimum degree at least $d n$ for some fixed $d > 0$, the distribution $G \cup \mathbb{G}(n,p)$ over the supergraphs of $G$ is referred to as a (random) {\sl perturbation} of $G$. We consider the distribution of edge-coloured graphs arising from assigning each edge of the random perturbation $G \cup \mathbb{G}(n,p)$ a colour, chosen independently and uniformly at random from a set of colours of size $r := r(n)$. We prove that such edge-coloured graph distributions a.a.s. admit rainbow Hamilton cycles whenever the edge-density of the random perturbation satisf…
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。
Near rainbow Hamilton cycles in dense graphs
针对Andersen提出的正常边着色图中近彩虹哈密顿圈问题，本文证明了满足全局n/8有界且最小度≥(1/2+ε)n的一般图存在含n-o(n)种颜色的哈密顿圈，并指出常数1/8最优。
A \emph{$k$--bisection} of a bridgeless cubic graph $G$ is a $2$--colouring of its vertex set such that the colour classes have the same cardinality and all connected components in the two subgraphs induced by the colour classes have order at most $k$. Ban and Linial conjectured that {\em every bridgeless cubic graph admits a $2$--bisection except for the Petersen graph}. In this note, we prove Ban--Linial's conjecture for claw--free cubic graphs.
The grid Ramsey number $ G(r) $ is the smallest number $ n $ such that every edge-colouring of the grid graph $Γ_{n,n} := K_n \times K_n$ with $r$ colours induces a rectangle whose parallel edges receive the same colour. We show $ G(r) \leq r^{\binom{r+1}{2}} - \left( 1/4 - o(1) \right) r^{\binom{r}{2}+1} $, slightly improving the currently best known upper bound due to Gyárfás.
Neumann-Lara and Škrekovski conjectured that every planar digraph is $2$-colourable. We show that this conjecture is equivalent to the more general statement that all oriented $K_5$-minor-free graphs are $2$-colourable.
The four-colour conjecture was brought to public attention in 1854, most probably by Francis or Frederick Guthrie. This moves back by six years the date of the earliest known publication.
Rainbow Hamiltonicity in uniformly coloured perturbed digraphs
本文证明了在边均匀着色的随机扰动有向图中，当底图最小半度满足条件时，以高概率存在彩虹有向哈密顿圈，改进了此前无向图设置下的相关结论。
A sequence is called non-repetitive if no of its subsequences forms a repetition (a sequence $r_1,r_2,\dots,r_{2n}$ such that $r_i=r_{n+i}$ for all $1\leq i \leq n$). Let $G$ be a graph whose vertices are coloured. A colouring $φ$ of the graph $G$ is non-repetitive if the sequence of colours on every path in $G$ is non-repetitive. The Thue chromatic number, denoted by $π(G)$, is the minimum number of colours of a non-repetitive colouring of $G$. In this short note we present a general upper bound for the Thue chromatic number for the lexicographic product $G\circ H$ of graphs $G$ and $H$ with …
An edge-coloured graph $G$ is rainbow connected if there exists a rainbow path between any two vertices. A graph $G$ is said to be $k$-rainbow connected if there exists an edge-colouring of $G$ with at most $k$ colours that is rainbow connected. For integers $n$ and $k$, let $t(n,k)$ denote the minimum number of edges in $k$-rainbow connected graphs of order $n$. In this note, we prove that $t(n,k) = \lceil k(n-2)/(k-1) \rceil$ for all $n, k \ge 3$
A note on multicolour Erdős-Hajnal conjecture
该研究将多色Erdős-Hajnal猜想归约为主团颜色数与禁用模式相同或多一种的情形，并发现带禁用模式的着色团中齐次集存在非单调性：主图多一种颜色可能减小最大齐次集的大小。
Enumerating Hamiltonian Paths and Cycles
本文证明了完全多部图中定向哈密顿路径与回路的计数公式，采用折叠技术关联普通路径图与合法着色路径图。
Let the edges of the complete graph $K_n$ be coloured red or blue, and let $G$ be a graph with $|V(G)| < n$. Then ot(n,G) is defined to be the minimum integer, if it exists, such that any such colouring of $K_n$ contains a copy of $G$ with $r$ red edges and $b$ blue edges for any $r,b \geq 0$ with $r+b= e(G)$. If ot(n,G) exists for every sufficiently large $n$, we say that $G$ is \emph{omnitonal}. Omnitonal graphs were introduced by Caro, Hansberg and Montejano [arXiv:1810.12375,2019]. Now let $G_1$, $G_2$ be two copies of $G$ with their edges coloured red or blue. If there is a colour-preserv…
The least $k$ admitting a proper edge colouring $c:E\to\{1,2,\ldots,k\}$ of a graph $G=(V,E)$ without isolated edges such that $\sum_{e\ni u}c(e)\neq \sum_{e\ni v}c(e)$ for every $uv\in E$ is denoted by $χ'_Σ(G)$. It has been conjectured that $χ'_Σ(G)\leq Δ+ 2$ for every connected graph of order at least three different from the cycle $C_5$, where $Δ$ is the maximum degree of $G$. It is known that $χ'_Σ(G) = Δ+ O(Δ^\frac{5}{6}\ln^\frac{1}{6}Δ)$ for a graph $G$ without isolated edges. We improve this upper bound to $χ'_Σ(G) = Δ+ O(Δ^\frac{1}{2})$ using a simpler approach involving a combinatori…
A note on interval colourings of graphs
本文研究图的区间着色厚度，证明了n顶点图的最大区间着色厚度介于Ω(log n/log log n)与n^{5/6+o(1)}之间，并据此回答了一个公开问题、否定了一个猜想，同时证实了平面图区间着色至多用3n/2-2种颜色的猜想。
A fractional colouring of a graph $G$ is a function that assigns a non-negative real value to all possible colour-classes of $G$ containing any vertex of $G$, such that the sum of these values is at least one for each vertex. The fractional chromatic number is the minimum sum of the values assigned by a fractional colouring over all possible such colourings of $G$. Introduced by Bosica and Tardif, fractional covers are an extension of fractional colourings whereby the real-valued function acts on all possible subgraphs of $G$ belonging to a given class of graphs. The fractional chromatic numbe…
The road colouring theorem characterizes the class of strongly connected directed graphs with constant out-degree that admit a synchronizing road colouring. The subject of this paper is a pair of related conjectures that generalize the road colouring theorem to graphs with non-constant out-degree; we give reasons to believe that both of these conjectures are true. Our main results focus on two classes of graphs, proving both conjectures for one class of graphs and one of the conjectures for an additional class of graphs. We also present computer simulations that give some empirical evidence fo…
In this note we show every orientation of a connected cubic graph admits an oriented 8-colouring. This lowers the best-known upper bound for the chromatic number of the family of orientations of connected cubic graphs. We further show that every such oriented graph admits a 2-dipath 7-colouring. These results imply that either the chromatic number for the family of oriented connected cubic graphs equals the 2-dipath chromatic number or the long-standing conjecture of Sopena [Journal of Graph Theory 25:191-205 1997] regarding the chromatic number of orientations of connected cubic graphs is fal…
Hadwiger's Conjecture states that every $K_{t+1}$-minor-free graph is $t$-colourable. It is widely considered to be one of the most important conjectures in graph theory. If every $K_{t+1}$-minor-free graph has minimum degree at most $δ$, then every $K_{t+1}$-minor-free graph is $(δ+1)$-colourable by a minimum-degree-greedy algorithm. The purpose of this note is to prove a slightly better upper bound.
Given a graph on $n$ vertices and an assignment of colours to the edges, a rainbow Hamilton cycle is a cycle of length $n$ visiting each vertex once and with pairwise different colours on the edges. Similarly (for even $n$) a rainbow perfect matching is a collection of $n/2$ independent edges with pairwise different colours. In this note we show that if we randomly colour the edges of a random geometric graph with sufficiently many colours, then a.a.s. the graph contains a rainbow perfect matching (rainbow Hamilton cycle) if and only if the minimum degree is at least $1$ (respectively, at leas…
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 $G$ has the Perfect-Matching-Hamiltonian property (PMH-property) if for each one of its perfect matchings, there is another perfect matching of $G$ such that the union of the two perfect matchings yields a Hamiltonian cycle of $G$. The study of graphs that have the PMH-property, initiated in the 1970s by Las Vergnas and Häggkvist, combines three well-studied properties of graphs, namely matchings, Hamiltonicity and edge-colourings. In this work, we study these concepts for cubic graphs in an attempt to characterise those cubic graphs for which every perfect matching corresponds to one …
A rainbow subgraph of an edge-coloured graph has all edges of distinct colours. A random d-regular graph with d even, and having edges coloured randomly with d/2 of each of n colours, has a rainbow Hamilton cycle with probability tending to 1 as n tends to infinity, provided d is at least 8.
In this note, we introduce and study a new version of neighbour-distinguishing arc-colourings of digraphs. An arc-colouring $γ$ of a digraph $D$ is proper if no two arcs with the same head or with the same tail are assigned the same colour. For each vertex $u$ of $D$, we denote by $S_γ^-(u)$ and $S_γ^+(u)$ the sets of colours that appear on the incoming arcs and on the outgoing arcs of $u$, respectively. An arc colouring $γ$ of $D$ is \emph{neighbour-distinguishing} if, for every two adjacent vertices $u$ and $v$ of $D$, the ordered pairs $(S_γ^-(u),S_γ^+(u))$ and $(S_γ^-(v),S_γ^+(v))$ are dis…
Layer barriers for colour-biased tight Hamilton cycles
本文针对均匀超图构造有色偏紧致哈密顿环的层状障碍，通过 $k=17$、$a=8$ 的反例及渐近分析，揭示其内部成员与边界构造的行为差异。
The size-Ramsey number $\hat{R}(F,r)$ of a graph $F$ is the smallest integer $m$ such that there exists a graph $G$ on $m$ edges with the property that any colouring of the edges of $G$ with $r$ colours yields a monochromatic copy of $F$. In this short note, we give an alternative proof of the recent result of Krivelevich that $\hat{R}(P_n,r) = O((\log r)r^2 n)$. This upper bound is nearly optimal, since it is also known that $\hat{R}(P_n,r) = Ω(r^2 n)$.
Colour-biased Hamilton cycles in dense graphs and random graphs
本文在稠密图和随机图中推广 Dirac 定理的偏差版本，渐近确定了最小度条件下 r-染色边集中单色出现次数的最大保证值 f_{r,α}(n)，并证明了相应的随机图结果。
A note on the equidistribution of $3$-colour partitions
本文证明了三族三色分拆的等分布结果，通过推导无穷乘积的渐近公式，获得了这些分拆族的渐近行为。
Let $G$ be a graph with maximum degree $Δ$ and without isolated vertices. An edge colouring $c$ of $G$ is conflict-free if the closed neighbourhood of every edge includes a uniquely coloured element. The least number of colours admitting such $c$ is the conflict-free chromatic index of $G$, denoted by $χ'_{CF}(G)$. In "Conflict-free chromatic number versus conflict-free chromatic index" [J. Graph Theory, 2022; 99: 349--358] it was recently proved by means of the probabilistic method that $χ'_{CF}(G)\leq C_1\log_2Δ+C_2$, where $C_1>337$ and $C_2$ are constants, whereas there are families of gra…
We study the multicolour discrepancy of spanning trees and Hamilton cycles in graphs. As our main result, we show that under very mild conditions, the $r$-colour spanning-tree discrepancy of a graph $G$ is equal, up to a constant, to the minimum $s$ such that $G$ can be separated into $r$ equal parts by deleting $s$ vertices. This result arguably resolves the question of estimating the spanning-tree discrepancy in essentially all graphs of interest. In particular, it allows us to immediately deduce as corollaries most of the results that appear in a recent paper of Balogh, Csaba, Jing and Pluh…
Colour diversity in spanning structures under Dirac-type conditions
本文研究了在Dirac型最小度条件下，生成结构中颜色多样性的结果，证明了在特定条件下存在具有大量不同颜色的哈密顿回路和排列。
Odd-Ramsey numbers of Hamilton cycles
本文研究哈密顿圈的奇拉姆齐数，通过有限域构造和奇偶切换组合框架证明其渐近阶为$\Theta(\sqrt{n})$，并初步探讨Dirac图中该类奇拉姆齐数的性质。
In 2017, Ron Aharoni made the following conjecture about rainbow cycles in edge-coloured graphs: If $G$ is an $n$-vertex graph whose edges are coloured with $n$ colours and each colour class has size at least $r$, then $G$ contains a rainbow cycle of length at most $\lceil \frac{n}{r} \rceil$. One motivation for studying Aharoni's conjecture is that it is a strengthening of the Caccetta-Häggkvist conjecture on digraphs from 1978. In this article, we present a survey of Aharoni's conjecture, including many recent partial results and related conjectures. We also present two new results. Our main…
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。
Gallai's colouring theorem states that if the edges of a complete graph are 3-coloured, with each colour class forming a connected (spanning) subgraph, then there is a triangle that has all 3 colours. What happens for more colours: if we $k$-colour the edges of the complete graph, with each colour class connected, how many of the $\binom{k}{3}$ triples of colours must appear as triangles? In this note we show that the `obvious' conjecture, namely that there are always at least $\binom{k-1}{2}$ triples, is not correct. We determine the minimum asymptotically. This answers a question of Johnson.…
If $G$ is a more than one tough graph on $n$ vertices with $δ\ge \frac{n}{2}-a$ for a given $a>0$ and $n$ is large enough then $G$ is hamiltonian.
A rainbow Dirac theorem for loose Hamilton cycles in hypergraphs
该研究证明了在满足特定度数阈值的k-均匀超图中，任何全局有界的着色都包含一个彩虹松散哈密顿回路。
In 1992, Manoussakis conjectured that a strongly 2-connected digraph $D$ on $n$ vertices is hamiltonian if for every two distinct pairs of independent vertices $x,y$ and $w,z$ we have $d(x)+d(y)+d(w)+d(z)\geq 4n-3$. In this note we show that $D$ has a Hamilton path, which gives an affirmative evidence supporting this conjecture.
We improve upon the lower bound for 3-colour hypergraph Ramsey numbers, showing, in the 3-uniform case, that \[r_3 (l,l,l) \geq 2^{l^{c \log \log l}}.\] The old bound, due to Erdős and Hajnal, was \[r_3 (l,l,l) \geq 2^{c l^2 \log^2 l}.\]
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