Tomasz Łuczak, Zahra Rahimi
We show that for every $η>0$, there exists $n_0$ such that for every even $n$, $n\ge n_0$, and every graph $G$ with $(2+η)n$ vertices and minimum degree at least $(7/4+4η)n$, each colouring of the edges of $G$ with three colours results in a monochromatic cycle of length $n$.
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We answer a question of Gyárfás and Sárközy from 2013 by showing that every 2-edge-coloured complete 3-uniform hypergraph can be partitioned into two monochromatic tight paths of different colours. We also give a lower bound for the number of tight paths needed to partition any 2-edge-coloured complete r-partite r-uniform hypergraph. Finally, we show that any 2-edge-coloured complete bipartite graph has a partition into a monochromatic cycle and a monochromatic path, of different colours, unless the colouring is a split colouring.
In 1959, Goodman determined the minimum number of monochromatic triangles in a complete graph whose edge set is two-coloured. Goodman also raised the question of proving analogous results for complete graphs whose edge sets are coloured with more than two colours. In this paper, we determine the minimum number of monochromatic triangles and the colourings which achieve this minimum in a sufficiently large three-coloured complete graph.
Confirming and extending a conjecture by Guggiari, we show that every countable $(r+1)$-edge-coloured complete symmetric digraph containing no directed paths of edge-length $\ell_i$ for any colour $i\leq r$ can be covered by $\prod_{i\leq r} \ell_i$ pairwise disjoint monochromatic directed paths in colour $r+1$.
Li, Nikiforov and Schelp conjectured that a 2-edge coloured graph G with order n and minimal degree strictly greater than 3n/4 contains a monochromatic cycle of length l, for all l at least four and at most n/2. We prove this conjecture for sufficiently large n and also find all 2-edge coloured graphs with minimal degree equal to 3n/4 that do not contain all such cycles. Finally we show that, for all positive constants d and sufficiently large n, a 2-edge coloured graph G of order n with minimal degree at least 3n/4 either contains a monochromatic cycle of length at least (2/3+d/2)n, or, in on…
We show that for any colouring of the edges of the complete bipartite graph $K_{n,n}$ with 3 colours there are 5 disjoint monochromatic cycles which together cover all but $o(n)$ of the vertices. In the same situation, 18 disjoint monochromatic cycles together cover all vertices.
We investigate the problem of determining how many monochromatic trees are necessary to cover the vertices of an edge-coloured random graph. More precisely, we show that for $p\gg n^{-1/6}{(\ln n)}^{1/6}$, in any $3$-edge-colouring of the random graph $G(n,p)$ we can find three monochromatic trees such that their union covers all vertices. This improves, for three colours, a result of Bucić, Korándi and Sudakov.
Monochromatic components in dense 2-edge-coloured balanced bipartite graphs
该研究针对稠密二边染色平衡二分图，证明满足最小度条件的图可被至多3个单色分支覆盖，并构造实例说明该最小度条件是最优的。
In 1978, Richard Rado showed that every edge-coloured complete graph of countably infinite order can be partitioned into monochromatic paths of different colours. He asked whether this remains true for uncountable complete graphs and a notion of \emph{generalised paths}. In 2016, Daniel Soukup answered this in the affirmative and conjectured that a similar result should hold for complete bipartite graphs with bipartition classes of the same infinite cardinality, namely that every such graph edge-coloured with $r$ colours can be partitioned into $2r-1$ monochromatic generalised paths with each …
We show that for every η > 0 there exists an integer n_0 such that every 2-colouring of the 3-uniform complete hypergraph on n \geq n_0 vertices contains two disjoint monochromatic tight cycles of distinct colours that together cover all but at most ηn vertices. The same result holds if we replace tight cycles with loose cycles.
Let $G = (V,E)$ be a simple graph and let $\{R,B\}$ be a partition of $E$. We prove that whenever $|E| + \min\{ |R|, |B| \} > { |V| \choose 2 }$, there exists a subgraph of $G$ isomorphic to $K_3$ which contains edges from both $R$ and $B$. We conjecture a natural generalization to partitions with more blocks.
We prove that for all integers $Δ,r \geq 2$, there is a constant $C = C(Δ,r) >0$ such that the following is true for every sequence $\mathcal{F} = \{F_1, F_2, \ldots\}$ of graphs with $v(F_n) = n$ and $Δ(F_n) \leq Δ$, for each $n \in \mathbb{N}$. In every $r$-edge-coloured $K_n$, there is a collection of at most $C$ monochromatic copies from $\mathcal{F}$ whose vertex-sets partition $V(K_n)$. This makes progress on a conjecture of Grinshpun and Sárközy.
Confirming a conjecture of Gyárfás, we prove that, for all natural numbers $k$ and $r$, the vertices of every $r$-edge-coloured complete $k$-uniform hypergraph can be partitioned into a bounded number (independent of the size of the hypergraph) of monochromatic tight cycles. We further prove that, for for all natural numbers $p$ and $r$, the vertices of every $r$-edge-coloured complete graph can be partitioned into a bounded number of $p$-th powers of cycles, settling a problem of Elekes, Soukup, Soukup and Szentmiklóssy. In fact we prove a common generalisation of both theorems which further …
Almost partitioning every $2$-edge-coloured complete $k$-graph into $k$ monochromatic tight cycles
研究红蓝着色完全$k$-图的单色紧圈划分问题，证明其中存在$k$个顶点互不相交的单色紧圈，可覆盖除$o(n)$外的全部顶点。
Balogh, Barát, Gerbner, Gyárfás, and Sárközy proposed the following conjecture. Let $G$ be a graph on $n$ vertices with minimum degree at least $3n/4$. Then for every $2$-edge-colouring of $G$, the vertex set $V(G)$ may be partitioned into two vertex-disjoint cycles, one of each colour. We prove that this conjecture holds for $n$ large enough, improving approximate results by the aforementioned authors and by DeBiasio and Nelsen.
Monochromatic cycle partitions of $r$-edge-coloured graphs with high minimum degree
该研究解决了高最小度数r边着色图的单色环覆盖问题，证明了所需单色环数量的上界，并推翻了关于单色树覆盖的猜想。
An open conjecture of Erdős states that for every positive integer $k$ there is a (least) positive integer $f(k)$ so that whenever a tournament has its edges colored with $k$ colors, there exists a set $S$ of at most $f(k)$ vertices so that every vertex has a monochromatic path to some point in $S$. We consider a related question and show that for every (finite or infinite) cardinal $κ>0$ there is a cardinal $ λ_κ$ such that in every $κ$-edge-coloured tournament there exist disjoint vertex sets $K,S$ with total size at most $ λ_κ$ so that every vertex $ v $ has a monochromatic path of length a…
For every $n\in\mathbb{N}$ and $k\geq2$, it is known that every $k$-edge-colouring of the complete graph on $n$ vertices contains a monochromatic connected component of order at least $\frac{n}{k-1}$. For $k\geq3$, it is known that the complete graph can be replaced by a graph $G$ with $δ(G)\geq(1-\varepsilon_k)n$ for some constant $\varepsilon_k$. In this paper, we show that the maximum possible value of $\varepsilon_3$ is $\frac16$. This disproves a conjecture of Gyárfas and Sárközy.
We show that for all $\ell, k, n$ with $\ell \leq k/2$ and $(k-\ell)$ dividing $n$ the following hypergraph-variant of Lehel's conjecture is true. Every $2$-edge-colouring of the $k$-uniform complete hypergraph $\mathcal{K}_n^{(k)}$ on $n$ vertices has at most two disjoint monochromatic $\ell$-cycles in different colours that together cover all but at most $4(k-\ell)$ vertices. If $\ell \leq k/3$, then at most two $\ell$-cycles cover all but at most $2(k-\ell)$ vertices. Furthermore, we can cover all vertices with at most $4$ ($3$ if $\ell\leq k/3$) disjoint monochromatic $\ell$-cycles.
Monochromatic partitions in 2-edge-coloured bipartite graphs
本研究针对Gyarfas--Lehel猜想，证明在随机二部图与高最小度二部图中，任意2-染色均可用至多三个顶点不相交的单色组件覆盖几乎所有顶点。
Let $K_{\mathbb{N}}$ be the complete symmetric digraph on the positive integers. Answering a question of DeBiasio and McKenney, we construct a $2$-colouring of the edges of $K_{\mathbb{N}}$ in which every monochromatic path has density~$0$. However, if we restrict the length of monochromatic paths in one colour, then no example as above can exist: We show that every $(r+1)$-edge-coloured complete symmetric digraph (of arbitrary infinite cardinality) containing no directed paths of edge-length $\ell_i$ for any colour $i\leq r$ can be covered by $\prod_{i\leq r} \ell_i$ pairwise disjoint monochr…
We consider a generalisation of the classical Ramsey theory setting to a setting where each of the edges of the underlying host graph is coloured with a {\em set} of colours (instead of just one colour). We give bounds for monochromatic tree covers in this setting, both for an underlying complete graph, and an underlying complete bipartite graph. We also discuss a generalisation of Ramsey numbers to our setting and propose some other new directions. Our results for tree covers in complete graphs imply that a stronger version of Ryser's conjecture holds for $k$-intersecting $r$-partite $r$-unif…
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.
Monochromatic odd cycles in edge-coloured complete graphs
针对Erdős与Graham提出的求最小L(q)问题，本文证明边着色完全图K_{2^q+1}中必含长度至多为O(2^q/q^{1-o(1)})的单色奇环，给出L(q)的首个非平凡上界。
The upper density of an infinite graph $G$ with $V(G) \subseteq \mathbb{N}$ is defined as $\overline{d}(G) = \limsup_{n \rightarrow \infty}{|V(G) \cap \{1,\ldots,n\}|}/{n}$. Let $K_{\mathbb{N}}$ be the infinite complete graph with vertex set $\mathbb{N}$. Corsten, DeBiasio, Lamaison and Lang showed that in every $2$-edge-colouring of $K_{\mathbb{N}}$, there exists a monochromatic path with upper density at least $(12 + \sqrt{8})/17$, which is best possible. In this paper, we extend this result to $k$-edge-colouring of $K_{\mathbb{N}}$ for $k \ge 3$. We conjecture that every $k$-edge-coloured $…
A conjecture of Erdős, Gyárfás, and Pyber says that in any edge-colouring of a complete graph with r colours, it is possible to cover all the vertices with r vertex-disjoint monochromatic cycles. So far, this conjecture has been proven only for r = 2. In this paper we show that in fact this conjecture is false for all r > 2. In contrast to this, we show that in any edge-colouring of a complete graph with three colours, it is possible to cover all the vertices with three vertex-disjoint monochromatic paths, proving a particular case of a conjecture due to Gyárfás. As an intermediate result we s…
P. Erdős proved that every 2-edge coloured complete graph on the natural numbers can be vertex decomposed into two monochromatic paths of different colour. This result was extended by R. Rado to an arbitrary finite number of colours. We prove that the vertices of every finite-edge coloured infinite complete graph can be partitioned into disjoint monochromatic paths of different colours. This answers a question of R. Rado from 1978.
Polynomial bounds for monochromatic tight cycle partition in $r$-edge-coloured $K_n^{(k)}$
针对 $r$-边染色完全 $k$-图的单色紧环顶点划分，本文将已有结果中塔型的循环数量常数改进为关于 $r$ 的多项式界。
Partitioning a 2-edge-coloured graph of minimum degree $2n/3 + o(n)$ into three monochromatic cycles
本文证明了最小度至少为 $2n/3 + o(n)$ 的红蓝边着色图可被划分为三个单色环，近似验证了 Pokrovskiy 的猜想且本质紧确。
We show that any complete $k$-partite graph $G$ on $n$ vertices, with $k \ge 3$, whose edges are two-coloured, can be covered with two vertex-disjoint monochromatic paths of distinct colours. We prove this under the necessary assumption that the largest partition class of $G$ contains at most $n/2$ vertices. This extends known results for complete and complete bipartite graphs. Secondly, we show that in the same situation, all but $o(n)$ vertices of the graph can be covered with two vertex-disjoint monochromatic cycles of distinct colours, if colourings close to a split colouring are excluded.…
Given an $r$-edge-coloured complete graph $K_n$, how many monochromatic connected components does one need in order to cover its vertex set? This natural question is a well-known essentially equivalent formulation of the classical Ryser's conjecture which, despite a lot of attention over the last 50 years, still remains open. A number of recent papers consider a sparse random analogue of this question, asking for the minimum number of monochromatic components needed to cover the vertex set of an $r$-edge-coloured random graph $\mathcal{G}(n,p)$. Recently, Bucić, Korándi and Sudakov established…
A classical result of Erdős, Gyárfás and Pyber states that any $r$-edge-coloured complete graph has a partition into $O(r^2 \log r)$ monochromatic cycles. Here we determine the minimum degree threshold for this property. More precisely, we show that there exists a constant $c$ such that any $r$-edge-coloured graph on $n$ vertices with minimum degree at least $n/2 + c \cdot r \log n$ has a partition into $O(r^2)$ monochromatic cycles. We also provide constructions showing that the minimum degree condition and the number of cycles are essentially tight.
Erdős, Gyárfás and Pyber showed that every $r$-edge-coloured complete graph $K_n$ can be covered by $25 r^2 \log r$ vertex-disjoint monochromatic cycles (independent of $n$). Here, we extend their result to the setting of binomial random graphs. That is, we show that if $p = p(n) = Ω(n^{-1/(2r)})$, then with high probability any $r$-edge-coloured $G(n,p)$ can be covered by at most $1000 r^4 \log r $ vertex-disjoint monochromatic cycles. This answers a question of Korándi, Mousset, Nenadov, Škorić and Sudakov.
Bounded diameter monochromatic component covers
本文证明了Ryser猜想与Milićević猜想等价，并展示了该猜想在r=5时成立，同时获得了关于非完全图推广猜想的新结果。
We (re-)prove that in every 3-edge-coloured tournament in which no vertex is incident with all colours there is either a cyclic rainbow triangle or a vertex dominating every other vertex monochromatically.
Rainbow subgraphs of star-coloured graphs
本文针对星着色图的彩虹子图问题，确定了不含给定图彩虹副本的大型完全图星着色的最大颜色数，拓展了Axenovich与Iverson在广义拉姆齐数相关问题上的结果。
Let $G$ be an edge-coloured graph. The minimum colour degree $δ^c(G)$ of $G$ is the largest integer $k$ such that, for every vertex $v$, there are at least $k$ distinct colours on edges incident to $v$. We say that $G$ is properly coloured if no two adjacent edges have the same colour. In this paper, we show that, for any $\varepsilon >0$ and $n$ large, every edge-coloured graph $G$ with $δ^c(G) \ge (1/2+\varepsilon)n$ contains a properly coloured cycle of length at least $\min\{ n , \lfloor 2 δ^c(G)/3 \rfloor\}$.
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 $n \ge n_0$…
Let $c$ be an edge-colouring of a graph $G$ such that for every vertex $v$ there are at least $d \ge 2$ different colours on edges incident to $v$. We prove that $G$ contains a properly coloured path of length 2d or a properly coloured cycle of length at least $d+1$. Moreover, if $G$ does not contain any properly coloured cycle, then there exists a properly coloured path of length $3 \times 2^{d-1}-2$.
Answering a question raised by Dudek and Prałat, we show that if $pn\rightarrow \infty$, w.h.p.,~whenever $G=G(n,p)$ is $2$-coloured, there exists a monochromatic path of length $n(2/3+o(1))$. This result is optimal in the sense that $2/3$ cannot be replaced by a larger constant. As part of the proof we obtain the following result which may be of independent interest. We show that given a graph $G$ on $n$ vertices with at least $(1-ε)\binom{n}{2}$ edges, whenever $G$ is $2$-edge-coloured, there is a monochromatic path of length at least $(2/3-100\sqrtε)n$. This is an extension of the classical…
Partitioning 2-edge-coloured bipartite graphs into monochromatic cycles
本文研究2-边着色完全平衡二部图划分为单色圈的问题，证明任意2-边着色的 $K_{n,n}$ 至多可划分为4个单色圈，改进了此前14个圈的上界。
Superlinear Lower Bounds for Monochromatic Path Partitions
本文以超线性形式否定Gyárfás的单色路径划分猜想，证明足够大r时r边染色完全图需至少(1-o(1))rloglogr个单色路径，其构造还扩展到平衡二分图场景，否定了Pokrovskiy的相关猜想。
Ore- and Pósa-type conditions for partitioning $2$-edge-coloured graphs into monochromatic cycles
本文研究2-边染色图的单色圈划分，提出较弱的度序列条件并证明近似版本，同时证明Ore型度条件可划分除少量顶点外的三个单色圈。
Extending a result of Rado to hypergraphs, we prove that for all $s, k, t \in \mathbb{N}$ with $k \geq t \geq 2$, the vertices of every $r = s(k-t+1)$-edge-coloured countably infinite complete $k$-graph can be partitioned into the cores of at most $s$ monochromatic $t$-tight Berge-paths of different colours. We further describe a construction showing that this result is best possible.
We show that there is a rational vector space $V$ such that, whenever $V$ is finitely coloured, there is an infinite set $X$ whose sumset $X+X$ is monochromatic. Our example is the rational vector space of dimension $\sup\{\aleph_0,2^{\aleph_0},2^{2^{\aleph_0}},\ldots\,\}$. This complements a result of Hindman, Leader and Strauss, who showed that the result does not hold for dimension below $\aleph_ω$. So our result is best possible under GCH.
A subgraph of an edge-coloured complete graph is called rainbow if all its edges have different colours. In 1980 Hahn conjectured that every properly edge-coloured complete graph $K_n$ has a rainbow Hamiltonian path. Although this conjecture turned out to be false, it was widely believed that such a colouring always contains a rainbow cycle of length almost $n$. In this paper, improving on several earlier results, we confirm this by proving that every properly edge-coloured $K_n$ has a rainbow cycle of length $n-O(n^{3/4})$. One of the main ingredients of our proof, which is of independent int…
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.
A well-known result by Haxell and Kohayakawa states that the vertices of an $r$-coloured complete graph can be partitioned into $r$ monochromatic connected subgraphs of distinct colours; this is a slightly weaker variant of a conjecture by Erdős, Pyber and Gyárfás that states that there exists a partition into $r-1$ monochromatic connected subgraphs. We consider a variant of this problem, where the complete graph is replaced by a graph with large minimum degree, and prove two conjectures of Bal and DeBiasio, for two and three colours.
In this paper we introduce mixed coloured permutation, permutations with certain coloured cycles, and study the enumerative properties of these combinatorial objects. We derive the generating function, closed forms, recursions and combinatorial identities for the counting sequence, mixed Stirling numbers of the first kind. In this comprehensive study we consider further the conditions on the length of the cycles, $r$-mixed Stirling numbers and the connection to Bell polynomials.
We extend results of Gyárfás and Füredi on the largest monochromatic component in $r$-colored complete $k$-uniform hypergraphs to the setting of random hypergraphs. We also study long monochromatic loose cycles in $r$-colored random hypergraphs. In particular, we obtain a random analog of a result of Gyárfás, Sárközy, and Szemerédi on the longest monochromatic loose cycle in $2$-colored complete $k$-uniform hypergraphs.
Let $A$ be a finite matrix with rational entries. We say that $A$ is {\it doubly image partition regular\/} if whenever the set ${\mathbb N}$ of positive integers is finitely coloured, there exists $\vec x$ such that the entries of $A\vec x$ are all the same colour (or {\it monochromatic\/}) and also, the entries of $\vec x$ are monochromatic. Which matrices are doubly image partition regular? More generally, we say that a pair of matrices $(A,B)$, where $A$ and $B$ have the same number of rows, is {\it doubly kernel partition regular\/} if whenever ${\mathbb N}$ is finitely coloured, there ex…
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…
Monochromatic cycle partitions in 3-mean edge-colourings
针对Conlon与Stein提出的边着色完全图划分为有限个单色圈的问题，证明顶点关联颜色平均数至多为3时该命题成立，解决了r=3的首个公开情况。
A proof of a conjecture of Erdős and Gyárfás on monochromatic path covers
本文证明了 Erdős 和 Gyárfás 关于单色路径覆盖的猜想：在 n 个顶点的 2-边染色完全图中，可用 √n 条同色单色路径覆盖所有顶点，对所有足够大的 n 成立。
Erdős and Pach (1983) introduced the natural degree-based generalisations of Ramsey numbers, where instead of seeking large monochromatic cliques in a $2$-edge coloured complete graph, we seek monochromatic subgraphs of high minimum or average degree. Here we expand the study of these so-called quasi-Ramsey numbers in a few ways, in particular, to multiple colours and to uniform hypergraphs. Quasi-Ramsey numbers are known to exhibit a certain unique phase transition and we show that this is also the case across the settings we consider. Our results depend on a density-biased notion of hypergra…
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。
A properly colored cycle (path) in an edge-colored graph is a cycle (path) with consecutive edges assigned distinct colors. A monochromatic triangle is a cycle of length $3$ with the edges assigned a same color. It is known that every edge-colored complete graph without containing monochromatic triangles always contains a properly colored Hamilton path. In this paper, we investigate the existence of properly colored cycles in edge-colored complete graphs when monochromatic triangles are forbidden. We obtain a vertex-pancyclic analogous result combined with a characterization of all the excepti…
Exact three-component covers in 2-coloured random bipartite graphs
本文解决了随机二部图中二色三组件猜想，证明当边概率远大于根号下对数n除以n时，任何红蓝边着色都能用至多三个单色连通组件覆盖所有顶点，方法基于公共邻域扩展引理。
We prove that for any $r\in \mathbb{N}$, there exists a constant $C_r$ such that the following is true. Let $\mathcal{F}=\{F_1,F_2,\dots\}$ be an infinite sequence of bipartite graphs such that $|V(F_i)|=i$ and $Δ(F_i)\leq Δ$ hold for all $i$. Then in any $r$-edge coloured complete graph $K_n$, there is a collection of at most $\exp(C_rΔ)$ monochromatic subgraphs, each of which is isomorphic to an element of $\mathcal{F}$, whose vertex sets partition $V(K_n)$. This proves a conjecture of Corsten and Mendonça in a strong form and generalizes results on the multicolour Ramsey numbers of bounded-…
The Hales-Jewett theorem for alphabet of size 3 states that whenever the Hales-Jewett cube [3]^n is r-coloured there is a monochromatic line (for n large). Conlon and Kamcev conjectured that, for any n, there is a 2-colouring of [3]^n for which there is no monochromatic line whose active coordinate set is an interval. In this note we disprove this conjecture.
Let $G$ be a graph whose edges are coloured with $k$ colours, and $\mathcal H=(H_1,\dots , H_k)$ be a $k$-tuple of graphs. A monochromatic $\mathcal H$-decomposition of $G$ is a partition of the edge set of $G$ such that each part is either a single edge or forms a monochromatic copy of $H_i$ in colour $i$, for some $1\le i\le k$. Let $ϕ_{k}(n,\mathcal H)$ be the smallest number $ϕ$, such that, for every order-$n$ graph and every $k$-edge-colouring, there is a monochromatic $\mathcal H$-decomposition with at most $ϕ$ elements. Extending the previous results of Liu and Sousa ["Monochromatic $K_…
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