Monochromatic Trees Research Articles

Covering random graphs with monochromatic trees

AbstractGiven an ‐edge‐colored complete graph , 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 ‐edge‐colored random graph . Recently, Bucić, Korándi, and Sudakov established a connection between this problem and a certain Helly‐type local to global question for hypergraphs raised about 30 years ago by Erdős, Hajnal, and Tuza. We identify a modified version of the hypergraph problem which controls the answer to the problem of covering random graphs with monochromatic components more precisely. To showcase the power of our approach, we essentially resolve the 3‐color case by showing that is a threshold at which point three monochromatic components are needed to cover all vertices of a 3‐edge‐colored random graph, answering a question posed by Kohayakawa, Mendonça, Mota, and Schülke. Our approach also allows us to determine the answer in the general ‐edge colored instance of the problem, up to lower order terms, around the point when it first becomes bounded, answering a question of Bucić, Korándi, and Sudakov.

Covering random graphs by monochromatic trees

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, Bucic, Korandi and Sudakov established a connection between this problem and a certain Helly-type local to global question for hypergraphs raised about 30 years ago by Erdős, Hajnal and Tuza. We identify a modified version of the hypergraph problem which controls the answer to the problem of covering random graphs with monochromatic components more precisely. To showcase the power of our approach, we essentially resolve the $3$-colour case by showing that $(\log n / n)^{1/4}$ is a threshold at which point three monochromatic components are needed to cover all vertices of a $3$-edge-coloured random graph, answering a question posed by Kohayakawa, Mendonca, Mota and Schulke. Our approach also allows us to determine the answer in the general $r$-edge coloured instance of the problem, up to lower order terms, around the point when it first becomes bounded, answering a question of Bucic, Korandi and Sudakov.

Covering Graphs by Monochromatic Trees and Helly-Type Results for Hypergraphs

How many monochromatic paths, cycles or general trees does one need to cover all vertices of a given r-edge-coloured graph G? These problems were introduced in the 1960s and were intensively studied by various researchers over the last 50 years. In this paper, we establish a connection between this problem and the following natural Helly-type question in hypergraphs. Roughly speaking, this question asks for the maximum number of vertices needed to cover all the edges of a hypergraph H if it is known that any collection of a few edges of H has a small cover. We obtain quite accurate bounds for the hypergraph problem and use them to give some unexpected answers to several questions about covering graphs by monochromatic trees raised and studied by Bal and DeBiasio, Kohayakawa, Mota and Schacht, Lang and Lo, and Cirão, Letzter and Sahasrabudhe.

Covering 3-Edge-Colored Random Graphs with Monochromatic Trees

We investigate the problem of determining how many monochromatic trees are necessary to cover the vertices of an edge-colored random graph. More precisely, we show that for $p\gg n^{-1/6}{(\ln n)}^...

Restricted Online Ramsey Numbers of Matchings and Trees

Consider a two-player game between players Builder and Painter. Painter begins the game by picking a coloring of the edges of $K_n$, which is hidden from Builder. In each round, Builder points to an edge and Painter reveals its color. Builder's goal is to locate a particular monochromatic structure in Painter's coloring by revealing the color of as few edges as possible. The fewest number of turns required for Builder to win this game is known as the restricted online Ramsey number. In this paper, we consider the situation where this "particular monochromatic structure" is a large matching or a large tree. We show that in any $t$-coloring of $E(K_n)$, Builder can locate a monochromatic matching on at least $\frac{n-t+1}{t+1}$ edges by revealing at most $O(n\log t)$ edges. We show also that in any $3$-coloring of $E(K_n)$, Builder can locate a monochromatic tree on at least $n/2$ vertices by revealing at most $5n$ edges.

Monochromatic Partitions In Local Edge Colorings

An edge coloring of a graph is a local r-coloring if the edges incident to any vertex are colored with at most r distinct colors. In this paper, generalizing our earlier work, we study the following problem. Given a family of graphs $$\mathcal {F} $$ (for example matchings, paths, cycles, powers of cycles and paths, connected subgraphs) and fixed positive integers s, r, at least how many vertices can be covered by the vertices of no more than s monochromatic members of $$\mathcal {F} $$ in every local r-coloring of $$K_n$$ . Several problems and results are presented. In particular, we prove the following two results. First, if n is sufficiently large then in any local r-coloring of the edges of $$K_n$$ , the vertex set can be partitioned by the vertices of at most r monochromatic trees, which is sharp for local r-colorings (unlike for ordinary r-colorings according to the Ryser conjecture). Second, we show that we can partition the vertex set with at most O(r log r) monochromatic cycles in every local r-coloring of $$K_n$$ . This answers a question of Conlon and Stein and slightly generalizes one of my favorite joint results with Endre (and with Gyarfas and Ruszinko).

Monochromatic trees in random tournaments

AbstractWe prove that, with high probability, in every 2-edge-colouring of the random tournament on n vertices there is a monochromatic copy of every oriented tree of order $O(n{\rm{/}}\sqrt {{\rm{log}} \ n} )$. This generalizes a result of the first, third and fourth authors, who proved the same statement for paths, and is tight up to a constant factor.

Monochromatic trees in random graphs

AbstractBal and DeBiasio [Partitioning random graphs into monochromatic components, Electron. J. Combin.24(2017), Paper 1.18] put forward a conjecture concerning the threshold for the following Ramsey-type property for graphsG: everyk-colouring of the edge set ofGyieldskpairwise vertex disjoint monochromatic trees that partition the whole vertex set ofG. We determine the threshold for this property for two colours.

Monochromatic tree covers and Ramsey numbers for set-coloured graphs

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 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-uniform hypergraphs: they have a transversal of size at most r−k. (Similar results have been obtained by Király et al., see below.) However, we also show that the bound r−k is not best possible in general.

Monochromatic trees in random graphs

Bal and DeBiasio [Partitioning random graphs into monochromatic components, Electron. J. Combin. 24 (2017), no. 1, Paper 1.18] put forward a conjecture concerning the threshold for the following Ramsey-type property for graphs G: every r-colouring of the edge set of G yields r pairwise vertex disjoint monochromatic trees that partition the whole vertex set of G. We determine the threshold for this property for two colours.

The (vertex-)monochromatic index of a graph

A tree T in an edge-colored (vertex-colored) graph H is called a monochromatic (vertex-monochromatic) tree if all the edges (internal vertices) of T have the same color. For $$S\subseteq V(H)$$S⊆V(H), a monochromatic (vertex-monochromatic) S-tree in H is a monochromatic (vertex-monochromatic) tree of H containing the vertices of S. For a connected graph G and a given integer k with $$2\le k\le |V(G)|$$2≤k≤|V(G)|, the k-monochromatic index$$mx_k(G)$$mxk(G) (k-monochromatic vertex-index$$mvx_k(G)$$mvxk(G)) of G is the maximum number of colors needed such that for each subset $$S\subseteq V(G)$$S⊆V(G) of k vertices, there exists a monochromatic (vertex-monochromatic) S-tree. For $$k=2$$k=2, Caro and Yuster showed that $$mc(G)=mx_2(G)=|E(G)|-|V(G)|+2$$mc(G)=mx2(G)=|E(G)|-|V(G)|+2 for many graphs, but it is not true in general. In this paper, we show that for $$k\ge 3$$kź3, $$mx_k(G)=|E(G)|-|V(G)|+2$$mxk(G)=|E(G)|-|V(G)|+2 holds for any connected graph G, completely determining the value. However, for the vertex-version $$mvx_k(G)$$mvxk(G) things will change tremendously. We show that for a given connected graph G, and a positive integer L with $$L\le |V(G)|$$L≤|V(G)|, to decide whether $$mvx_k(G)\ge L$$mvxk(G)źL is NP-complete for each integer k such that $$2\le k\le |V(G)|$$2≤k≤|V(G)|. Finally, we obtain some Nordhaus---Gaddum-type results for the k-monochromatic vertex-index.

Partitioning 3-Edge-Colored Complete Equi-Bipartite Graphs by Monochromatic Trees under a Color Degree Condition

The monochromatic tree partition number of an $r$-edge-colored graph $G$, denoted by $t_r(G)$, is the minimum integer $k$ such that whenever the edges of $G$ are colored with $r$ colors, the vertices of $G$ can be covered by at most $k$ vertex-disjoint monochromatic trees. In general, to determine this number is very difficult. For 2-edge-colored complete multipartite graph, Kaneko, Kano, and Suzuki gave the exact value of $t_2(K(n_1,n_2,\cdots,n_k))$. In this paper, we prove that if $n\geq 3$, and $K(n,n)$ is 3-edge-colored such that every vertex has color degree 3, then $t_3(K(n,n))=3$.

Vertex partitions of r-edge-colored graphs

Let G be an edge-colored graph. The monochromatic tree partition problem is to find the minimum number of vertex disjoint monochromatic trees to cover the all vertices of G. In the authors' previous work, it has been proved that the problem is NP-complete and there does not exist any constant factor approximation algorithm for it unless P = NP. In this paper the authors show that for any fixed integer r � 5, if the edges of a graph G are colored by r colors, called an r-edge-colored graph, the problem remains NP-complete. Similar result holds for the monochromatic path (cycle) partition problem. Therefore, to find some classes of interesting graphs for which the problem can be solved in polynomial time seems interesting. A linear time algorithm for the monochromatic path partition problem for edge-colored trees is given.

Partitioning 2-edge-colored complete multipartite graphs into monochromatic cycles, paths and trees

In this paper we consider the problem of partitioning complete multipartite graphs with edges colored by 2 colors into the minimum number of vertex disjoint monochromatic cycles, paths and trees, respectively. For general graphs we simply address the decision version of these three problems the 2-PGMC, 2-PGMP and 2-PGMT problems, respectively. We show that both 2-PGMC and 2-PGMP problems are NP-complete for complete multipartite graphs and the 2-PGMT problem is NP-complete for bipartite graphs. This also implies that all these three problems are NP-complete for general graphs, which solves a question proposed by the authors in a previous paper. Nevertheless, we show that the 2-PGMT problem can be solved in polynomial time for complete multipartite graphs.

Partitioning complete multipartite graphs by monochromatic trees

The tree partition number of an r-edge-colored graph G, denoted by tr(G), is the minimum number k such that whenever the edges of G are colored with r colors, the vertices of G can be covered by at most k vertex-disjoint monochromatic trees. We determine t2(K(n1, n2,…, nk)) of the complete k-partite graph K(n1, n2,…, nk). In particular, we prove that t2(K(n, m)) = ⌊ (m-2)/2n⌋ + 2, where 1 ≤ n ≤ m. © 2004 Wiley Periodicals, Inc. J Graph Theory 48: 133–141, 2005

Special monochromatic trees in two‐colored complete graphs

For a positive integer k, a set of k + 1 vertices in a graph is a k-cluster if the difference between degrees of any two of its vertices is at most k − 1. Given any tree T with at least k3 edges, we show that for each graph G of sufficiently large order, either G or its complement contains a copy of T such that some vertices in the copy form a k-cluster in G. The same conclusion holds for any tree T having a vertex of degree more than k. © 1997 John Wiley & Sons, Inc.

Special monochromatic trees in two-colored complete graphs

Special monochromatic trees in two-colored complete graphs

Partitioning by Monochromatic Trees

Any r -edge-coloured n -vertex complete graph K n contains at most r monochromatic trees, all of different colours, whose vertex sets partition the vertex set of K n , provided n ⩾3 r 4 r ! (1−1/ r ) 3(1− r ) log r . This comes close to proving, for large n , a conjecture of Erdős, Gyárfás, and Pyber, which states that r −1 trees suffice for all n .