Monochromatic Paths Research Articles

Flashes and Rainbows in Tournaments

Colour the edges of the complete graph with vertex set {1,2,⋯,n}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\{1, 2, \\dotsc , n\\}}$$\\end{document} with an arbitrary number of colours. What is the smallest integer f(l, k) such that if n>f(l,k)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n > f(l,k)$$\\end{document} then there must exist a monotone monochromatic path of length l or a monotone rainbow path of length k? Lefmann, Rödl, and Thomas conjectured in 1992 that f(l,k)=lk-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(l, k) = l^{k - 1}$$\\end{document} and proved this for l⩾(3k)2k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$l \\geqslant (3 k)^{2 k}$$\\end{document}. We prove the conjecture for l⩾k3(logk)1+o(1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$l \\geqslant k^3 (\\log k)^{1 + o(1)}$$\\end{document} and establish the general upper bound f(l,k)⩽k(logk)1+o(1)·lk-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(l, k) \\leqslant k (\\log k)^{1 + o(1)} \\cdot l^{k - 1}$$\\end{document}. This reduces the gap between the best lower and upper bounds from exponential to polynomial in k. We also generalise some of these results to the tournament setting.

Constraining MC-numbers by the connectivity of complement graphs

An edge-coloring of a graph G is called a monochromatic connection coloring (or MC-coloring for short) if any two vertices of G are joined by a monochromatic path (a path whose edges are colored with the same color). The monochromatic connection number (or MC-number for short) of G, denoted by mc(G), is the maximum number of colors used in an MC-coloring of G. In this paper we main explore the relationships between mc(G) and the connectivity of its complement graph G‾. We give a method for computing mc(G) if G‾ is disconnected and give the sharp upper bounds of mc(G) when G‾ is k-connected, where 1≤k≤3. If G is a graph with G‾k-connected and k≥4, then mc(G)=m−n+2 (this result has been proved by Caro and Yuster) and we further talk about the characteristics of MC-colorings of G. In addition, we prove that any graph G with mc(G)≥m−n+t contains a Kt.

Colouring bottomless rectangles and arborescences

We study problems related to colouring families of bottomless rectangles in the plane, in an attempt to improve the polychromatic k-colouring numbermk⁎. This number is the smallest m such that any collection of bottomless rectangles can be k-coloured so that any m-fold covered point is covered by all k colours. We show that for many families of bottomless rectangles, such as unit-width bottomless rectangles, or bottomless rectangles whose left corners lie on a line, mk⁎ is linear in k. We present the lower bound mk⁎≥2k−1 for general families.We also investigate semi-online colouring algorithms, which need not colour each vertex immediately, but must maintain a proper colouring. We prove that for many sweeping orders, for any positive integers m,k, there is no semi-online algorithm that can k-colour bottomless rectangles presented in that order, so that any m-fold covered point is covered by at least two colours. This holds even for translates of quadrants, and is a corollary of a stronger result for arborescence colourings: Any semi-online colouring algorithm that colours an arborescence presented in post-order may produce arbitrarily long monochromatic paths.

Monochromatic Paths in 2-Edge-Coloured Graphs and Hypergraphs


 
 
 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 k-partite k-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.
 
 

Upper density of monochromatic paths in edge-coloured infinite complete graphs and bipartite graphs

The upper density of an infinite graph G with V(G)⊆N is defined as d¯(G)=lim supn→∞|V(G)∩{1,…,n}|/n. Let KN be the infinite complete graph with vertex set N. Corsten, DeBiasio, Lamaison and Lang showed that in every 2-edge-colouring of KN, there exists a monochromatic path with upper density at least (12+8)/17, which is best possible. In this paper, we extend this result to k-edge-colouring of KN for k≥3. We conjecture that every k-edge-coloured KN contains a monochromatic path with upper density at least 1/(k−1), which is best possible (when k−1 is a prime power). We prove that this is true when k=3 and asymptotically when k=4. Furthermore, we show that this problem can be deduced from its bipartite variant, which is of independent interest.

An improvement on Łuczak's connected matchings method

A connected matching in a graph G $G$ is a matching contained in a connected component of G $G$ . A well-known method due to Łuczak reduces problems about monochromatic paths and cycles in complete graphs to problems about monochromatic connected matchings in almost complete graphs. We show that these can be further reduced to problems about monochromatic connected matchings in complete graphs. We illustrate the potential of this new reduction by showing how it can be used to determine the 3-colour Ramsey number of long paths, using a simpler argument than the original one by Gyárfás, Ruszinkó, Sárközy, and Szemerédi (2007).

Proper vertex-pancyclicity of edge-colored complete graphs without monochromatic paths of length three

Let Gc be an edge-colored graph of order n. The minimum color degree δc(Gc) of Gc is the minimum number of different colors appearing on the edges incident with a vertex of Gc. A cycle in Gc is proper if no two adjacent edges are assigned the same color. The graph Gc is called properly vertex-pancyclic if each vertex of Gc is contained in a proper cycle of length k for every k with 3≤k≤n. We show in this paper that if δc(Knc)≥n+12 and Knc contains no monochromatic P4, then Knc is properly vertex-pancyclic. Our result partially solves a conjecture proposed by Fujita and Magnant which says that, if δc(Knc)≥n+12, then Knc is properly vertex-pancyclic.

New Lower Bounds on the Size-Ramsey Number of a Path

We prove that for all graphs with at most $(3.75-o(1))n$ edges there exists a 2-coloring of the edges such that every monochromatic path has order less than $n$. This was previously known to be true for graphs with at most $2.5n-7.5$ edges. We also improve on the best-known lower bounds in the $r$-color case.

$(k,H)$-kernels in nearly tournaments

Let $H$ be a digraph possibly with loops, $D$ a digraph without loops, and $\rho : A(D) \rightarrow V(H)$ a coloring of $A(D)$ ($D$ is said to be an $H$-colored digraph). If $W=(x_{0}, \ldots , x_{n})$ is a walk in $D$, and $i \in \{ 0, \ldots , n-1 \}$, we say that there is an obstruction on $x_{i}$ whenever $(\rho(x_{i-1}, x_{i}), \rho (x_{i}, x_{i+1})) \notin A(H)$ (when $x_{0} = x_{n}$ the indices are taken modulo $n$). We denote by $O_{H}(W)$ the set $\{ i \in \{0, \ldots , n-1 \} :$ there is an obstruction on $x_{i} \}$. The $H$-length of $W$, denoted by $l_{H}(W)$, is defined by $|O_{H}(W)|+1$ whenever $x_{0} \neq x_{n}$, or $|O_{H}(W)|$ in other case. A $(k, H)$-kernel of an $H$-colored digraph $D$ ($k \geq 2$) is a subset of vertices of $D$, say $S$, such that, for every pair of different vertices in $S$, every path between them has $H$-length at least $k$, and for every vertex $x \in V(D) \setminus S$ there exists an $xS$-path with $H$-length at most $k-1$. This concept widely generalize previous nice concepts as kernel, $k$-kernel, kernel by monochromatic paths, kernel by properly colored paths, and $H$-kernel. In this paper, we will study the existence of $(k,H)$-kernels in interesting classes of digraphs, called nearly tournaments, which have been large and widely studied due its applications and theoretical results. We will show several conditions that guarantee the existence of $(k,H)$-kernel in tournaments, $r$-transitive digraphs, $r$-quasi-transitive digraphs, multipartite tournaments, and local tournaments.

Restricted domination in Quasi-transitive and 3-Quasi-transitive digraphs

Let H be a digraph possibly with loops and D a digraph without loops with a coloring of its arcs c:A(D)→V(H) (D is said to be an H-colored digraph). A directed path W in D is said to be an H-path if and only if the consecutive colors encountered on W form a directed walk in H. A subset N of vertices of D is said to be an H-kernel if (1) for every pair of different vertices in N there is no H-path between them and (2) for every vertex u in V(D)∖N there exists an H-path in D from u to N. Under this definition an H-kernel is a kernel whenever A(H)=0̸; an H-kernel is a kernel by monochromatic paths whenever A(H)={(u,u):u∈V(H)}; an H-kernel is a properly colored kernel whenever H has no loops and an H-kernel is a kernel by rainbow paths whenever H has no directed cycles.Let k be an integer, with k≥2. D is k-quasi-transitive if for every pair of vertices u and v of D, the existence of a directed path of length k from u to v implies that {(u,v),(v,u)}∩A(D)≠0̸.In Galeana-Sánchez and O’Reilly-Regueiro (2013), in Delgado-Escalante et al. (2018) and in Galeana-Sánchez and Rojas-Monroy (2006) the authors establish sufficient conditions to guarantee the existence of kernels by monochromatic paths in 3-quasi-transitive digraphs, the existence of properly colored kernels in 2-quasi-transitive digraphs and the existence of kernels in 2-quasi-transitive digraphs, respectively. In this paper we investigate the problem of the existence of H-kernels in quasi-transitive and in 3-quasi-transitive digraphs by means of a partition of V(H).We give some examples which show that each hypothesis in the main result for quasi-transitive digraphs is tight.

Monochromatic paths and cycles in 2-edge-coloured graphs with large minimum degree

AbstractA graph G arrows a graph H if in every 2-edge-colouring of G there exists a monochromatic copy of H. Schelp had the idea that if the complete graph $K_n$ arrows a small graph H, then every ‘dense’ subgraph of $K_n$ also arrows H, and he outlined some problems in this direction. Our main result is in this spirit. We prove that for every sufficiently large n, if $n = 3t+r$ where $r \in \{0,1,2\}$ and G is an n-vertex graph with $\delta(G) \ge (3n-1)/4$ , then for every 2-edge-colouring of G, either there are cycles of every length $\{3, 4, 5, \dots, 2t+r\}$ of the same colour, or there are cycles of every even length $\{4, 6, 8, \dots, 2t+2\}$ of the samecolour.Our result is tight in the sense that no longer cycles (of length $>2t+r$ ) can be guaranteed and the minimum degree condition cannot be reduced. It also implies the conjecture of Schelp that for every sufficiently large n, every $(3t-1)$ -vertex graph G with minimum degree larger than $3|V(G)|/4$ arrows the path $P_{2n}$ with 2n vertices. Moreover, it implies for sufficiently large n the conjecture by Benevides, Łuczak, Scott, Skokan and White that for $n=3t+r$ where $r \in \{0,1,2\}$ and every n-vertex graph G with $\delta(G) \ge 3n/4$ , in each 2-edge-colouring of G there exists a monochromatic cycle of length at least $2t+r$ .

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.

Color computational ghost imaging based on a generative adversarial network.

A novel, to the best of our knowledge, color computational ghost imaging scheme is presented for the reconstruction of a color object image, which greatly simplifies the experimental setup and shortens the acquisition time. Compared to conventional schemes, it only adopts one digital light projector to project color speckles and one single-pixel detector to receive the light intensity, instead of utilizing three monochromatic paths separately and synthesizing the three branch results. Severe noise and color distortion, which are common in ghost imaging, can be removed by the utilization of a generative adversarial network, because it has advantages in restoring the image's texture details and generating the image's match to a human's subjective feelings over other generative models in deep learning. The final results can perform consistently better visual quality with more realistic and natural textures, even at the low sampling rate of 0.05.

Rainbow Monochromatic k-Edge-Connection Colorings of Graphs

A path in an edge-colored graph is called a monochromatic path if all edges of the path have a same color. We call k paths \(P_1,\ldots ,P_k\) rainbow monochromatic paths if every \(P_i\) is monochromatic and for any two \(i\ne j\), \(P_i\) and \(P_j\) have different colors. An edge-coloring of a graph G is said to be a rainbow monochromatic k-edge-connection coloring (or \(RMC_k\)-coloring for short) if every two distinct vertices of G are connected by at least k rainbow monochromatic paths. We use \(rmc_k(G)\) to denote the maximum number of colors that ensures G has an \(RMC_k\)-coloring, and this number is called the rainbow monochromatic k-edge-connection number. We prove the existence of \(RMC_k\)-colorings of graphs, and then give some bounds of \(rmc_k(G)\) and present some graphs whose \(rmc_k(G)\) reaches the lower bound. We also obtain the threshold function for \(rmc_k(G(n,p))\ge f(n)\), where \(\left\lfloor \frac{n}{2}\right\rfloor > k\ge 1\).

Multicolorful connectivity of trees

In 2011, Caro and Yuster introduced the colorful monochromatic connectivity of graphs, which considers the maximum number of colors used in an edge-coloring of a graph such that for any two vertices, there is a monochromatic path joining them. In this paper, we replace the condition monochromatic path with the path with at most k colors and determine the color number for trees.

Determining the Circular Flow Number of a Cubic Graph

A circular nowhere-zero $r$-flow on a bridgeless graph $G$ is an orientation of the edges and an assignment of real values from $[1, r-1]$ to the edges in such a way that the sum of incoming values equals the sum of outgoing values for every vertex. The circular flow number, $\phi_c(G)$, of $G$ is the infimum over all values $r$ such that $G$ admits a nowhere-zero $r$-flow. A flow has its underlying orientation. If we subtract the number of incoming and the number of outgoing edges for each vertex, we get a mapping $V(G) \to \mathbb{Z}$, which is its underlying balanced valuation. In this paper we describe efficient and practical polynomial algorithms to turn balanced valuations and orientations into circular nowhere zero $r$-flows they underlie with minimal $r$. Using this algorithm one can determine the circular flow number of a graph by enumerating balanced valuations. For cubic graphs we present an algorithm that determines $\phi_c(G)$ in case that $\phi_c(G) \leqslant 5$ in time $O(2^{0.6\cdot|V(G)|})$. If $\phi_c(G) > 5$, then the algorithm determines that $\phi_c(G) > 5$ and thus the graph is a counterexample to Tutte's $5$-flow conjecture. The key part is a procedure that generates all (not necessarily proper) $2$-vertex-colourings without a monochromatic path on three vertices in $O(2^{0.6\cdot|V(G)|})$ time. We also prove that there is at most $2^{0.6\cdot|V(G)|}$ of them.

Extremal graphs and classification of planar graphs by MC-numbers

An edge-coloring of a connected graph $G$ is called a {\em monochromatic connection coloring} (MC-coloring for short) if any two vertices of $G$ are connected by a monochromatic path in $G$. For a connected graph $G$, the {\em monochromatic connection number} (MC-number for short) of $G$, denoted by $mc(G)$, is the maximum number of colors that ensure $G$ has a monochromatic connection coloring by using this number of colors. This concept was introduced by Caro and Yuster in 2011. They proved that $mc(G)\leq m-n+k$ if $G$ is not a $k$-connected graph. In this paper we depict all graphs with $mc(G)=m-n+k+1$ and $mc(G)= m-n+k$ if $G$ is a $k$-connected but not $(k+1)$-connected graph. We also prove that $mc(G)\leq m-n+4$ if $G$ is a planar graph, and classify all planar graphs by their monochromatic connectivity numbers.

Partitioning edge-coloured infinite complete bipartite graphs into monochromatic paths

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 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 colour being used at most twice.In the present paper, we give an affirmative answer to Soukup’s conjecture.

Extremal graphs with maximum monochromatic connectivity

An edge-coloring of a connected graph is called a monochromatic connection coloring (MC-coloring, for short) if there is a monochromatic path joining each two vertices of the graph. For a connected graph G, the monochromatic connection number of G, denoted by mc(G), is defined to be the maximum number of colors allowed in an MC-coloring of G. This concept was introduced by Caro and Yuster (2011). They proved that |E(G)|−|V(G)|+2≤mc(G)≤|E(G)|−|V(G)|+χ(G) for any connected graph G. Some graphs achieving the lower bound have been determined. In this paper, we first give a new upper bound for mc(G), which is characterized by the structures of extremal colorings. Using this new bound, we then characterize all connected graphs G achieving the upper bound mc(G)=|E(G)|−|V(G)|+χ(G).

Extensions of Richardson’s theorem for infinite digraphs and (𝒜, ℬ)-kernels

Let D be a digraph and and two subsets of where = {P: P is a non trivial finite path in D}. A subset N of V(D) is said to be an ()-kernel of D if: (1) for every {u,v} N there exists no uv-path P such that P (N is -independent), (2) for every vertex x in V(D) there exist y in N and P in such that P is an xy-path (N is -absorbent). As a particular case, the concept of ()-kernel generalizes the concept of kernel when = = A(D). A classical result in kernel theory is Richardson’s theorem which establishes that if D is a finite digraph without odd cycles, then D has a kernel. In this paper, the original results are sufficient conditions for the existence of ()-kernels in possibly infinite digraphs, in particular we will present some generalizations of Richardson’s theorem for infinite digraphs. Also we will deduce some conditions for the existence of kernels by monochromatic paths, H-kernels and (k,l)-kernels in possibly infinite digraphs.