Edge-colored Complete Graphs Research Articles

Cycles of length 3 and 4 in edge-colored complete graphs with restrictions in the color transitions

Let G be a graph and H a graph possibly with loops. We will say that a graph G is an H-colored graph if and only if there exists a function c:E(G)⟶V(H)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$c:E(G)\\longrightarrow V(H)$$\\end{document}. A cycle (v1,…,vk,v1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(v_1,\\ldots ,v_k,v_1)$$\\end{document} is an H-cycle if and only if (c(v1v2),…,c(vk-1vk),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(c(v_1 v_2),\\ldots ,c(v_{k-1}v_k),$$\\end{document}c(vkv1),c(v1v2))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$c(v_kv_1), c(v_1 v_2))$$\\end{document} is a walk in H. Whenever H is a complete graph without loops, an H-cycle is a properly colored cycle. In this paper, we work with an H-colored complete graph, namely G, with local restrictions given by an auxiliary graph, and we show sufficient conditions implying that every vertex in V(G) is contained in an H-cycle of length 3 (respectively 4). As a consequence, we obtain some well-known results in the theory of properly colored walks.

On the existence of cycles with restrictions in the color transitions in edge-colored complete graphs

Consider the following edge-coloring of a graph G. Let H be a graph possibly with loops, an H-coloring of a graph G is defined as a function c:E(G)→V(H).\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$c : E(G) \\rightarrow V(H).$$\\end{document} We will say that G is an H-colored graph whenever we are taking a fixed H-coloring of G. A cycle (x0,x1,…,xn,x0),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(x_0,x_1,\\ldots ,x_n,x_0),$$\\end{document} in an H-colored graph, is an H-cycle if and only if (c(x0x1),c(x1x2),…,c(xnx0),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(c(x_0x_1),c(x_1x_2),\\ldots , c(x_nx_0),$$\\end{document}c(x0x1))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$c(x_0x_1))$$\\end{document} is a walk in H. Notice that the graph H determines what color transitions are allowed in a cycle in order to be an H-cycle, in particular, when H is a complete graph without loops, every H-cycle is a properly colored cycle. In this paper, we give conditions on an H-colored complete graph G, with local restrictions, implying that every vertex of G is contained in an H-cycle of length at least 5. As a consequence, we obtain a previous result about properly colored cycles. Finally, we show an infinite family of H-colored complete graphs fulfilling the conditions of the main theorem, where the graph H is not a complete k-partite graph for any k in N.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {N}}.$$\\end{document}

Colored Unavoidable Patterns and Balanceable Graphs

We study a Turán-type problem on edge-colored complete graphs. We show that for any $r$ and $t$, any sufficiently large $r$-edge-colored complete graph on $n$ vertices with $\Omega(n^{2-1/tr^r})$ edges in each color contains a member from certain finite family $\mathcal{F}_t^r$ of $r$-edge-colored complete graphs. We conjecture that $\Omega(n^{2-1/t})$ edges in each color are sufficient to find a member from ${\mathcal{F}}_t^r$. A result of Girão and Narayanan confirms this conjecture when $r=2$. Next, we study a related problem where the corresponding Turán threshold is linear. We call an edge-coloring of a path $P_{rk}$ balanced if each color appears $k$ times in the coloring. We show that any $3$-edge-coloring of a large complete graph with $kn+o(n)$ edges in each color contains a balanced $P_{3k}$. This is tight up to a constant factor of $2$. For more colors, the problem becomes surprisingly more delicate. Already for $r=7$, we show that even $n^{2-o(1)}$ edges from each color does not guarantee existence of a balanced $P_{7k}$.

Edge-disjoint properly colored cycles in edge-colored complete graphs

In an edge-colored graph G, let dmon(v) denote the maximum number of edges with the same color incident with a vertex v in G, called the monochromatic-degree of v. The maximum value of dmon(v) over all vertices v∈V(G) is called the maximum monochromatic-degree of G, denoted by Δmon(G). Li et al. in 2019 conjectured that every edge-colored complete graph G of order n with Δmon(G)≤n−3k+1 contains k vertex-disjoint properly colored (PC for short) cycles of length at most 4, and they showed that the conjecture holds for k=2. Han et al. showed that every edge-colored complete graph G of order n with Δmon(G)≤n−2k contains k PC cycles of different lengths. They further got the condition Δmon(G)≤n−6 for the existence of two vertex-disjoint PC cycles of different lengths. In this paper, we consider the problems of the existence of edge-disjoint PC cycles of length at most 4 (different lengths) in an edge-colored complete graph G of order n.

Parallel Connectivity in Edge-Colored Complete Graphs: Complexity Results

Parallel Connectivity in Edge-Colored Complete Graphs: Complexity Results

Proper Cycles and Rainbow Cycles in 2-triangle-free edge-colored Complete Graphs

Proper Cycles and Rainbow Cycles in 2-triangle-free edge-colored Complete Graphs

Properly colored cycles of different lengths in edge-colored complete graphs

A cycle in an edge-colored graph is called properly colored if all of its adjacent edges have distinct colors. Let Knc be an edge-colored complete graph with n vertices and let k be a positive integer. Denote by Δmon(Knc) the maximum number of edges of the same color incident with a vertex of Knc. In this paper, we show that (i) if Δmon(Knc)⩽n−2k, then Knc contains k properly colored cycles of different lengths and the bound is sharp; (ii) if Δmon(Knc)⩽n−2k+1−2k+4, then Knc contains k vertex-disjoint properly colored cycles of different lengths; in particular, Δmon(Knc)⩽n−6 suffices for the existence of two vertex-disjoint properly colored cycles of different lengths.

Rainbow disjoint union of clique and matching in edge-colored complete graph

Rainbow disjoint union of clique and matching in edge-colored complete graph

Monochromatic-degree conditions for properly colored cycles in edge-colored complete graphs

Let G be an edge-colored graph and v be a vertex of G. Define the monochromatic-degree dmon(v) of v to be the maximum number of edges with the same color incident with v in G, and the maximum monochromatic-degree Δmon(G) of G to be the maximum value of dmon(v) over all vertices v of G. A cycle (path) in G is called properly colored if any two adjacent edges of the cycle (path) have distinct colors. Wang et al. in 2014 showed that an edge-colored complete graph Knc with Δmon(Knc)<⌊n2⌋ contains a properly colored cycle of length at least ⌈n2⌉+2. In this paper, we obtain a generalization of their result that an edge-colored complete graph Knc of order n with Δmon(Knc)=d≤n−2 contains a properly colored cycle of length at least n−d+1.

Disjoint properly colored cycles in edge-colored complete bipartite graphs

Let Kn,m be an edge-colored complete bipartite graph with 1≤n≤m and Δmon(Kn,m) be the number of the leaves of a maximum monochromatic star in Kn,m. In this paper, we show that if Δmon(Kn,m)≤n−2k+1, then Kn,m has k disjoint properly colored cycles.

Properly colored 2-factors of edge-colored complete bipartite graphs

Let Gc be an edge-colored graph. The minimum color degree of Gc, denoted δc(G), is the largest integer k such that for every vertex v, there are at least k distinct colors on edges incident with v. A cycle in Gc is called properly colored if any of its adjacent edges have distinct colors. A properly colored 2-factor of Gc is a spanning subgraph consisting of vertex-disjoint properly colored cycles of Gc. It was conjectured in the literature that, if δc(Km,n)≥m+n4+1, then each vertex of Km,nc is contained in a properly colored cycle of length l for any even integer l with 4≤l≤min⁡{2m,2n}. In this paper, we consider the existence of properly colored 2-factor in edge-colored bipartite graphs and obtain the following two results: (i) if δc(Kn,n)>3n/4, then Kn,nc contains a properly colored 2-factor. (ii) For every number ε with 0<ε<1/2, there exists an integer n0(ε) such that every Kn,nc with n≥n0(ε) and δc(Kn,n)≥(1/2+ε)n contains a properly colored 2-factor.

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.

Rainbow Perfect and Near-Perfect Matchings in Complete Graphs with Edges Colored by Circular Distance

Given an edge-colored complete graph Kn on n vertices, a perfect (respectively, near-perfect) matching M in Kn with an even (respectively, odd) number of vertices is rainbow if all edges have distinct colors. In this paper, we consider an edge coloring of Kn by circular distance, and we denote the resulting complete graph by K●n. We show that when K●n has an even number of vertices, it contains a rainbow perfect matching if and only if n=8k or n=8k+2, where k is a nonnegative integer. In the case of an odd number of vertices, Kirkman matching is known to be a rainbow near-perfect matching in K●n. However, real-world applications sometimes require multiple rainbow near-perfect matchings. We propose a method for using a recursive algorithm to generate multiple rainbow near-perfect matchings in K●n.

Color neighborhood union conditions for proper edge-pancyclicity of edge-colored complete graphs

Let G be an edge-colored complete graph on n vertices such that there exist at least n distinct colors on edges incident to every pair of its vertices. In this paper, we first show that every edge of G with n≥6k−19 is contained in a properly colored cycle of length k. Further, we prove that if G contains no monochromatic triangles, then there exists a properly colored path of length l for every 1≤l≤n−1 between each pair of vertices of G and every vertex of G is contained in a properly colored cycle of length k for any 3≤k≤n.

Properly colored cycles in edge-colored complete graphs without monochromatic triangle: A vertex-pancyclic analogous result

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 monochromatic triangle 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 exceptions. As a consequence, we partially confirm a structural conjecture given by Bollobás and Erdős (1976) and an algorithmic conjecture given by Gutin and Kim (2009).

On the number of alternating paths in random graphs

In the noisy channel model from coding theory, we wish to detect errors introduced during transmission by optimizing various parameters of the code. Bennett, Dudek, and LaForge framed a variation of this problem in the language of alternating paths in edge-colored complete bipartite graphs in 2016. Here, we extend this problem to the random graph G(n,p). We seek the alternating connectivity, κr,ℓ(G), which is the maximum t such that there is an r-edge-coloring of G such that any pair of vertices is connected by t internally disjoint and alternating (i.e. no consecutive edges of the same color) paths of length ℓ. We have three main results about how this parameter behaves in G(n,p) that basically cover all ranges of p: one for paths of length two, one for the dense case, and one for the sparse case. For paths of length two, we found that κr,ℓ(G) is essentially the codegree of a pair of vertices. For the dense case when p is constant, we were able to achieve the natural upper bounds of minimum degree (minus some intersection) or the total number of disjoint paths between a pair of vertices. For the sparse case, we were able to find colorings that achieved the natural obstructions of minimum degree or (in a slightly less precise result) the total number of paths of a certain length in a graph. We broke up this sparse case into ranges of p corresponding to when G(n,p) has diameter k or k+1. We close with some remarks about a similar parameter and a generalization to pseudorandom graphs.

Edge-Colored Complete Graphs Containing No Properly Colored Odd Cycles

It is well known that a graph is bipartite if and only if it contains no odd cycles. Gallai characterized edge colorings of complete graphs containing no properly colored triangles in recursive sense. In this paper, we completely characterize edge-colored complete graphs containing no properly colored odd cycles and give an efficient algorithm with complexity $$O(n^{3})$$ for deciding the existence of properly colored odd cycles in an edge-colored complete graph of order n. Moreover, we show that for two integers k, m with $$m\geqslant k\geqslant 3$$ , where $$k-1$$ and m are relatively prime, an edge-colored complete graph contains a properly colored cycle of length $$\ell \equiv k\ (\text{mod}\ m)$$ if and only if it contains a properly cycle of length $$\ell '\equiv k\ (\text{mod }\ m)$$ , where $$\ell '< 2m^{2}(k-1)+3m$$ .

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.

Some algorithmic results for finding compatible spanning circuits in edge-colored graphs

A compatible spanning circuit in a (not necessarily properly) edge-colored graph G is a closed trail containing all vertices of G in which any two consecutively traversed edges have distinct colors. Sufficient conditions for the existence of extremal compatible spanning circuits (i.e., compatible Hamilton cycles and Euler tours), and polynomial-time algorithms for finding compatible Euler tours have been considered in previous literature. More recently, sufficient conditions for the existence of more general compatible spanning circuits in specific edge-colored graphs have been established. In this paper, we consider the existence of (more general) compatible spanning circuits from an algorithmic perspective. We first show that determining whether an edge-colored connected graph contains a compatible spanning circuit is an NP-complete problem. Next, we describe two polynomial-time algorithms for finding compatible spanning circuits in edge-colored complete graphs. These results in some sense give partial support to a conjecture on the existence of compatible Hamilton cycles in edge-colored complete graphs due to Bollobás and Erdős from the 1970s.

On characterizing the critical graphs for matching Ramsey numbers

Given simple graphs H1,H2,…,Hc, the Ramsey number r(H1,H2,…,Hc) is the smallest positive integer n such that every edge-colored Kn with c colors contains a subgraph in color i isomorphic to Hi for some i∈{1,2,…,c}. The critical graphs for r(H1,H2,…,Hc) are edge-colored complete graphs on r(H1,H2,…,Hc)−1 vertices with c colors which contain no subgraphs in color i isomorphic to Hi for any i∈{1,2,…,c}. For n1≥n2≥⋯≥nc≥1, Cockayne and Lorimer (1975) showed that r(n1K2,n2K2,…,ncK2)=n1+1+∑i=1c(ni−1), in which niK2 is a matching of size ni. Using the Gallai–Edmonds Theorem, we characterized all the critical graphs for r(n1K2,n2K2,…,ncK2), implying a new proof for this Ramsey number.