Edge-colored Graph Research Articles

Complexity results for two kinds of colored disconnections of graphs

The concept of rainbow disconnection number of graphs was introduced by Chartrand et al. (2018). Inspired by this concept, we put forward the concepts of rainbow vertex-disconnection and proper disconnection in graphs. In this paper, we first show that it is NP-complete to decide whether a given edge-colored graph G has a proper edge-cut separating two specified vertices, even though the graph G has \(\Delta (G)=4\) or is bipartite. Then, for a graph G with \(\Delta (G)\le 3\) we show that \(pd(G)\le 2\) and distinguish the graphs with \(pd(G)=1\) and 2, respectively. We also show that it is NP-complete to decide whether a given vertex-colored graph G is rainbow vertex-disconnected, even though the graph G has \(\Delta (G)=3\) or is bipartite.

Conflict-free Connection Number and Independence Number of a Graph

An edge-colored graph G is conflict-free connected if any two of its vertices are connected by a path, which contains a color used on exactly one of its edges. The conflict-free connection number of a connected graph G, denoted by cfc(G), is defined as the minimum number of colors that are required in order to make G conflict-free connected. In this paper, we investigate the relation between the conflict-free connection number and the independence number of a graph. We firstly show that cfc(G) ≤ α(G) for any connected graph G, and give an example to show that the bound is sharp. With this result, we prove that if T is a tree with \(\Delta(T)\geq\frac{\alpha(T)+2}{2}\), then cfc(T) = Δ(T).

On the existence and non-existence of improper homomorphisms of oriented and $2$-edge-coloured graphs to reflexive targets

We consider non-trivial homomorphisms to reflexive oriented graphs in which some pair of adjacent vertices have the same image. Using a notion of convexity for oriented graphs, we study those oriented graphs that do not admit such homomorphisms. We fully classify those oriented graphs with tree-width $2$ that do not admit such homomorphisms and show that it is NP-complete to decide if a graph admits an orientation that does not admit such homomorphisms. We prove analogous results for $2$-edge-coloured graphs. We apply our results on oriented graphs to provide a new tool in the study of chromatic number of orientations of planar graphs -- a long-standing open problem.

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 \(\left\lfloor \frac{n}{2}\right\rfloor > k\ge 1\).

Anti-Ramsey Number of Triangles in Complete Multipartite Graphs

An edge-colored graph is called rainbow if all its edges are colored distinct. The anti-Ramsey number of a graph family $${\mathcal {F}}$$ in the graph G, denoted by $$AR{(G,{\mathcal {F}})}$$ , is the maximum number of colors in an edge-coloring of G without rainbow subgraph in $${\mathcal {F}}$$ . The anti-Ramsey number for the short cycle $$C_3$$ has been determined in a few graphs. Its anti-Ramsey number in the complete graph can be easily obtained from the lexical edge-coloring. Gorgol considered the problem in complete split graphs which contains complete graphs as a subclass. In this paper, we study the problem in the complete multipartite graph which further enlarges the family of complete split graphs. The anti-Ramsey numbers for $$C_3$$ and $$C_3^{+}$$ in complete multipartite graphs are determined. These results contain the known results for $$C_3$$ and $$C_3^{+}$$ in complete and complete split graphs as corollaries.

MỘT ĐIỀU KIỆN MỚI ĐỂ MỘT ĐỒ THỊ CÓ SỐ LIÊN THÔNG KHÔNG XUNG ĐỘT LÀ 2

A path in an edge-coloured graph is called conflict-free if there is a colour used on exactly one of its edges. An edge-coloured graph is said to be conflict-free connected if any two distinct vertices of the graph are connected by a conflict-free path. The conflict-free connection number, denoted by cf c(G), is the smallest number of colours needed in order to make G conflict-free connected. In this paper, we give a new condition to show that a connected non-complete graph G having cf c(G) = 2. This is an extension of a result by Chang et al. [1].

Upper bounds for the [formula omitted]-numbers and characterization of extremal graphs

For an edge-colored graph G, we call an edge-cut M of G monochromatic if the edges of M are colored with the same color. The graph G is called monochromatic disconnected if any two distinct vertices of G are separated by a monochromatic edge-cut. For a connected graph G, the monochromatic disconnection number (or MD-number for short) of G, denoted by md(G), is the maximum number of colors that are allowed in order to make G monochromatic disconnected. For graphs with diameter one, they are complete graphs and so their MD-numbers are 1. For graphs with diameter at least 3, we can construct 2-connected graphs such that their MD-numbers can be arbitrarily large; whereas for graphs G with diameter two, we show that if G is a 2-connected graph then md(G)≤2, and if G has a cut-vertex then md(G) is equal to the number of blocks of G. So, we will focus on studying 2-connected graphs with diameter two, and give two upper bounds of their MD-numbers depending on their connectivity and independent numbers, respectively. We also characterize the n2-connected graphs (with large connectivity) whose MD-numbers are 2 and the 2-connected graphs (with small connectivity) whose MD-numbers achieve the upper bound n2 (these graphs are called extremal graphs). For graphs with connectivity less than n2, we show that if the connectivity of a graph is linear in its order n, then its MD-number is upper bounded by a constant, and this suggests us to leave a conjecture that for a k-connected graph G, md(G)≤nk.

On odd rainbow cycles in edge-colored graphs

Let G=(V,E) be an n-vertex edge-colored graph. In 2013, H. Li proved that if every vertex v∈V is incident to at least (n+1)∕2 distinctly colored edges, then G admits a rainbow triangle. We prove that the same hypothesis ensures a rainbow ℓ-cycle Cℓ whenever n≥432ℓ. This result is sharp for all odd integers ℓ≥3, and extends earlier work of the authors for when ℓ is even.

Proper vertex-pancyclicity of edge-colored complete graphs without joint monochromatic triangles

In an edge-colored graph (G,c), let dc(v) denote the number of colors on the edges incident with a vertex v of G and δc(G) denote the minimum value of dc(v) over all vertices v∈V(G). A cycle of (G,c) is called proper if any two adjacent edges of the cycle have distinct colors. An edge-colored graph (G,c) on n≥3 vertices is called properly vertex-pancyclic if each vertex of (G,c) is contained in a proper cycle of length ℓ for every ℓ with 3≤ℓ≤n. Fujita and Magnant conjectured that every edge-colored complete graph on n≥3 vertices with δc(G)≥n+12 is properly vertex-pancyclic. Chen, Huang and Yuan partially solve this conjecture by adding an extra condition that (G,c) does not contain any monochromatic triangle. In this paper, we show that this conjecture is true if the edge-colored complete graph contain no joint monochromatic triangles.

On reduced complexity of closed piecewise linear 5-manifolds

The goal of this paper is to give some theorems which relate to the problem of classifying combinatorial (resp. smooth) closed 5-manifolds up to piecewise-linear (PL) homeomorphism. For this, we use the combinatorial approach to the topology of PL manifolds by means of a special kind of edge-colored graphs, called crystallizations. Within this representation theory, Bracho and Montejano introduced in 1987 a nonnegative numerical invariant, called the reduced complexity, for any closed n-dimensional PL manifold. Here we obtain the complete classification of all closed connected smooth 5-manifolds of reduced complexity less than or equal to 20. In particular, this gives a combinatorial characterization of $$\mathbb {S}^2 \times \mathbb {S}^3$$ among closed connected spin PL 5-manifolds.

On Aharoni’s rainbow generalization of the Caccetta–Häggkvist conjecture

For a digraph G and v∈V(G), let δ+(v) be the number of out-neighbors of v in G. The Caccetta–Häggkvist conjecture states that for all k≥1, if G is a digraph with n=|V(G)| such that δ+(v)≥k for all v∈V(G), then G contains a directed cycle of length at most ⌈n∕k⌉. In Aharoni et al. (2019), Aharoni proposes a generalization of this conjecture, that a simple edge-colored graph on n vertices with n color classes, each of size k, has a rainbow cycle of length at most ⌈n∕k⌉. In this paper, we prove this conjecture if each color class has size Ω(klogk).

Strong rainbow disconnection in graphs

Let G be a nontrivial edge-colored connected graph. A rainbow edge-cut is an edge-cut R of G, and all edges of R have different colors in G. For two different vertices u and v of G, a u-v-edge-cut is an edge-cut separating them. An edge-colored graph G is called strong rainbow disconnected if for every two distinct vertices u and v of G, there exists a both rainbow and minimum u-v-edge-cut in G, and such an edge-coloring is called a strong rainbow disconnection coloring (srd-coloring for short) of G. For a connected graph G, the strong rainbow disconnection number (srd-number for short) of G, denoted by srd(G), is the minimum number of colors required to make G strong rainbow disconnected. In this paper, we first characterize the graphs with m edges satisfing srd(G) = k for each k ∈ {1, 2, m}, respectively, and we also show that the srd-number of a nontrivial connected graph G is equal to the maximum srd-number in the blocks of G. Secondly, we study the srd-numbers for the complete k-partite graphs, k-edge-connected k-regular graphs and grid graphs. Finally, we prove that for a connected graph G, computing srd(G) is NP-hard. In particular, we prove that it is NP-complete to decide if srd(G) = 3 for a connected cubic graph. We also show that the following problem is NP-complete: given an edge-colored (with an unbounded number of colors) connected graph G, check whether the given coloring makes G strong rainbow disconnected.

Smaller universal targets for homomorphisms of edge-colored graphs

For a graph G, the density of G, denoted D(G), is the maximum ratio of the number of edges to the number of vertices ranging over all subgraphs of G. For a class $$\mathcal {F}$$ of graphs, the value $$D(\mathcal {F})$$ is the supremum of densities of graphs in $$\mathcal {F}$$ . A k-edge-colored graph is a finite, simple graph with edges labeled by numbers $$1,\ldots ,k$$ . A function from the vertex set of one k-edge-colored graph to another is a homomorphism if the endpoints of any edge are mapped to two different vertices connected by an edge of the same color. Given a class $$\mathcal {F}$$ of graphs, a k-edge-colored graph $$\mathbb {H}$$ (not necessarily with the underlying graph in $$\mathcal {F}$$ ) is k-universal for $$\mathcal {F}$$ when any k-edge-colored graph with the underlying graph in $$\mathcal {F}$$ admits a homomorphism to $$\mathbb {H}$$ . Such graphs are known to exist exactly for classes $$\mathcal {F}$$ of graphs with acyclic chromatic number bounded by a constant. The minimum number of vertices in a k-uniform graph for a class $$\mathcal {F}$$ is known to be $$\Omega (k^{D(\mathcal {F})})$$ and $$O(k^{{\left\lceil D(\mathcal {F}) \right\rceil }})$$ . In this paper we close the gap by improving the upper bound to $$O(k^{D(\mathcal {F})})$$ for any rational $$D(\mathcal {F})$$ .

Proper Disconnection of Graphs

For an edge-colored graph G, a set F of edges of G is called a proper edge-cut if F is an edge-cut of G and any pair of adjacent edges in F are assigned different colors. An edge-colored graph is proper disconnected if for each pair of distinct vertices of G there exists a proper edge-cut separating them. For a connected graph G, the proper disconnection number of G, denoted by pd(G), is the minimum number of colors that are needed in order to make G proper disconnected. In this paper, we first give the exact values of the proper disconnection numbers for some special families of graphs. Next, we obtain a sharp upper bound of pd(G) for a connected graph G of order n, i.e, \(pd(G)\le \min \{ \chi '(G)-1, \left\lceil \frac{n}{2} \right\rceil \}\). Finally, we show that for given integers k and n, the minimum size of a connected graph G of order n with \(pd(G)=k\) is \(n-1\) for \(k=1\) and \(n+2k-4\) for \(2\le k\le \lceil \frac{n}{2}\rceil \).

Monochromatic Components in Edge-Coloured Graphs with Large Minimum Degree

For every $n\in\mathbb{N}$ and $k\geqslant2$, 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 $k\geqslant3$, it is known that the complete graph can be replaced by a graph $G$ with $\delta(G)\geqslant(1-\varepsilon_k)n$ 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.

Some results on strong conflict-free connection number of graphs

A path in an edge-colored graph is called a conflict-free path if there exists a color used on only one of its edges. An edge-colored graph is called conflict-free connected if there is a conflict-free path between each pair of distinct vertices. We call the graph [Formula: see text] strongly conflict-free connectedif there exists a conflict-free path of length [Formula: see text] for every two vertices [Formula: see text]. The strong conflict-free connection number of a connected graph [Formula: see text], denoted by [Formula: see text], is defined as the smallest number of colors that are required to make [Formula: see text] strongly conflict-free connected. The authors in [M. Ji, X. Li and X. Zhu, (Strong) Conflict-free connectivity: Algorithm and complexity, Theor. Comput. Sci. 804 (2020) 72–80] showed that the problem of deciding whether [Formula: see text] [Formula: see text] for a given graph [Formula: see text] is NP-complete. In this paper, we give mainly some bounds of strong conflict-free connection number of graphs and illustrate the difference between [Formula: see text] and [Formula: see text],[Formula: see text].

Rainbow Hamilton Cycles in Randomly Colored Randomly Perturbed Dense Graphs

Given an $n$-vertex graph $H$ with minimum degree at least $d n$ for some fixed $d > 0$, the distribution $H \cup \mathbb{G}(n,p)$ over the supergraphs of $H$ is referred to as a (random) perturbation of $H$. We consider the distribution of edge-colored graphs arising from assigning each edge of the random perturbation $H \cup \mathbb{G}(n,p)$ a color, chosen independently and uniformly at random from a set of colors of size $r := r(n)$. We prove that edge-colored graphs which are generated in this manner asymptotically almost surely admit rainbow Hamilton cycles whenever the edge-density of the random perturbation satisfies $p := p(n) \geq C/n$ for some fixed $C > 0$ and $r = (1 + o(1))n$. The number of colors used is clearly asymptotically best possible. In particular, this improves on a recent result of Anastos and Frieze [J. Graph Theory, 92 (2019), pp. 405--414] in this regard. As an intermediate result, which may be of independent interest, we prove that randomly edge-colored sparse pseudorandom graphs asymptotically almost surely admit an almost spanning rainbow path.

Implications in rainbow forbidden subgraphs

An edge-colored graph G is rainbow if each edge receives a different color. For a connected graph H, G is rainbow H-free if G does not contain a rainbow subgraph which is isomorphic to H. By definition, if H′ is a connected subgraph of H, every rainbow H′-free graph is rainbow H-free. In this note, we consider a kind of reverse implication, where H′ is a connected proper supergraph and every rainbow H′-free graph edge-colored in sufficiently many colors is rainbow H-free. We determine when this implication occurs.

On the anti-Ramsey number of forests

We call a subgraph of an edge-colored graph rainbow, if all of its edges have different colors. The anti-Ramsey number of a graph G in a complete graph Kn, denoted by ar(Kn,G), is the maximum number of colors in an edge-coloring of Kn with no rainbow copy of G. In this paper, we determine the exact value of the anti-Ramsey number for star forests and double stars and the approximate value of the anti-Ramsey number for linear forests.

Generalizations of the Ruzsa–Szemerédi and rainbow Turán problems for cliques

AbstractConsidering a natural generalization of the Ruzsa–Szemerédi problem, we prove that for any fixed positive integers r, s with r < s, there are graphs on n vertices containing $n^{r}e^{-O\left(\sqrt{\log{n}}\right)}=n^{r-o(1)}$ copies of Ks such that any Kr is contained in at most one Ks. We also give bounds for the generalized rainbow Turán problem ex (n, H, rainbow - F) when F is complete. In particular, we answer a question of Gerbner, Mészáros, Methuku and Palmer, showing that there are properly edge-coloured graphs on n vertices with $n^{r-1-o(1)}$ copies of Kr such that no Kr is rainbow.