Minimum Color Degree Research Articles

Counting rainbow triangles in edge‐colored graphs

AbstractLet be an edge‐colored graph on vertices. The minimum color degree of , denoted by , is defined as the minimum number of colors assigned to the edges incident to a vertex in . In 2013, Li proved that an edge‐colored graph on vertices contains a rainbow triangle if . In this paper, we obtain several estimates on the number of rainbow triangles through one given vertex in . As a consequence, we prove counting results for rainbow triangles in edge‐colored graphs. One main theorem states that the number of rainbow triangles in is at least , which is best possible by considering the rainbow ‐partite Turán graph, where its order is divisible by . This means that there are rainbow triangles in if , and rainbow triangles in if when . Both results are tight in the sense of the order of the magnitude. We also prove a counting version of a previous theorem on rainbow triangles under a color neighborhood union condition due to Broersma et al., and an asymptotically tight color degree condition forcing a colored friendship subgraph (i.e., rainbow triangles sharing a common vertex).

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.

Note on rainbow cycles in edge-colored graphs

Let G be a graph of order n with an edge-coloring c, and let δc(G) denote the minimum color-degree of G. A subgraph F of G is called rainbow if all edges of F have pairwise distinct colors. There have been a lot of results on rainbow cycles of edge-colored graphs. In this paper, we show that (i) if δc(G)>2n−13, then every vertex of G is contained in a rainbow triangle; (ii) if δc(G)>2n−13 and n≥13, then every vertex of G is contained in a rainbow C4; (iii) if G is complete, n≥7k−17 and δc(G)>n−12+k, then G contains a rainbow cycle of length at least k, where k≥5.

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.

Properly colored short cycles in edge-colored graphs

Properly colored cycles in edge-colored graphs are closely related to directed cycles in oriented graphs. As an analogy of the well-known Caccetta–Häggkvist Conjecture, we study the existence of properly colored cycles of bounded length in an edge-colored graph. We first prove that for all integers s and t with t≥s≥2, every edge-colored graph G with no properly colored Ks,t contains a spanning subgraph H which admits an orientation D such that every directed cycle in D is a properly colored cycle in G. Using this result, we show that for r≥4, if the Caccetta–Häggkvist Conjecture holds , then every edge-colored graph of order n with minimum color degree at least n/r+2n+1 contains a properly colored cycle of length at most r. In addition, we also obtain an asymptotically tight total color degree condition which ensures a properly colored (or rainbow) Ks,t.

Vertex-disjoint rainbow triangles in edge-colored graphs

Let G be an edge-colored graph of order n. The minimum color degree of G, denoted by δc(G), is the largest integer k such that for every vertex v, there are at least k distinct colors on edges incident to v. We say that an edge-colored graph is rainbow if all its edges have different colors. In this paper, we consider vertex-disjoint rainbow triangles in edge-colored graphs. Li (2013) showed that if δc(G)≥(n+1)∕2, then G contains a rainbow triangle and the lower bound is tight. Motivated by this result, we prove that if n≥20 and δc(G)≥(n+2)∕2, then G contains two vertex-disjoint rainbow triangles. In particular, we conjecture that if δc(G)≥(n+k)∕2, then G contains k vertex-disjoint rainbow triangles. For any integer k≥2, we show that if n≥16k−12 and δc(G)≥n∕2+k−1, then G contains k vertex-disjoint rainbow triangles. Moreover, we provide sufficient conditions for the existence of k edge-disjoint rainbow triangles.

Rainbow Matchings of size $m$ in Graphs with Total Color Degree at least $2mn$

The existence of a rainbow matching given a minimum color degree, proper coloring, or triangle-free host graph has been studied extensively. This paper generalizes these problems to edge colored graphs with given total color degree. In particular, we find that if a graph $G$ has total color degree $2mn$ and satisfies some other properties, then $G$ contains a matching of size $m$. These other properties include $G$ being triangle-free, $C_4$-free, properly colored, or large enough.

Color degree sum conditions for properly colored spanning trees in edge-colored graphs

For a vertex v of an edge-colored graph, the color degree of v is the number of colors appeared in edges incident with v. An edge-colored graph is called properly colored if no two adjacent edges have the same color. In this paper, we prove that if the minimum color degree sum of two adjacent vertices of an edge-colored connected graph G is at least |G|, then G has a properly colored spanning tree. This is a generalization of the result proved by Cheng, Kano and Wang. We also show the sharpness of this lower bound of the color degree sum.

Properly colored spanning trees in edge-colored graphs

A subgraph H of an edge-colored graph G is called a properly colored subgraph if no two adjacent edges of H have the same color, and is called a rainbow subgraph if no two edges of H have the same color. For a vertex v of G, the color degree of v, denoted by dGc(v), is the number of distinct colors appeared in the edges incident with v. Let δc(G) be the minimum value among the color degrees of all the vertices in G. We prove the following two theorems and show that the conditions on the minimum color degree are sharp. Let G be an edge-colored graph. If δc(G)≥|G|∕2, then G has a properly colored spanning tree. Moreover, if δc(G)≥|G|∕2 and the set of edges colored with any fixed color forms a subgraph of order at most (|G|∕2)+1, then G has a rainbow spanning tree.

Decomposing edge-colored graphs under color degree constraints

For an edge-colored graph G, the minimum color degree of G means the minimum number of colors on edges which are incident to each vertex of G. We prove that if G is an edge-colored graph with minimum color degree at least 5 then V (G) can be partitioned into two parts such that each part induces a subgraph with minimum color degree at least 2. We show this theorem by proving a much stronger form. Moreover, we point out an important relationship between our theorem and Bermond-Thomassen's conjecture in digraphs.

Color degree and monochromatic degree conditions for short properly colored cycles in edge‐colored graphs

AbstractFor an edge‐colored graph, its minimum color degree is defined as the minimum number of colors appearing on the edges incident to a vertex and its maximum monochromatic degree is defined as the maximum number of edges incident to a vertex with a same color. A cycle is called properly colored if every two of its adjacent edges have distinct colors. In this article, we first give a minimum color degree condition for the existence of properly colored cycles, then obtain the minimum color degree condition for an edge‐colored complete graph to contain properly colored triangles. Afterwards, we characterize the structure of an edge‐colored complete bipartite graph without containing properly colored cycles of length 4 and give the minimum color degree and maximum monochromatic degree conditions for an edge‐colored complete bipartite graph to contain properly colored cycles of length 4, and those passing through a given vertex or edge, respectively.

Cycle extension in edge-colored complete graphs

Let G be an edge-colored graph. The minimum color degree of G is the minimum number of different colors appearing on the edges incident with the vertices of G. In this paper, we study the existence of properly edge-colored cycles in (not necessarily properly) edge-colored complete graphs. Fujita and Magnant (2011) conjectured that in an edge-colored complete graph on n vertices with minimum color degree at least (n+1)∕2, each vertex is contained in a properly edge-colored cycle of length k, for all k with 3≤k≤n. They confirmed the conjecture for k=3 and k=4, and they showed that each vertex is contained in a properly edge-colored cycle of length at least 5 when n≥13, but even the existence of properly edge-colored Hamilton cycles is unknown (in complete graphs that satisfy the conditions of the conjecture). We prove a cycle extension result that implies that each vertex is contained in a properly edge-colored cycle of length at least the minimum color degree.

Rainbow cycles in edge-colored graphs

Let G be a graph of order n with an edge coloring c, and let δc(G) denote the minimum color degree of G, i.e., the largest integer such that each vertex of G is incident with at least δc(G) edges having pairwise distinct colors. A subgraph F⊂G is rainbow if all edges of F have pairwise distinct colors. In this paper, we prove that (i) if G is triangle-free and δc(G)>n3+1, then G contains a rainbow C4, and (ii) if δc(G)>n2+2, then G contains a rainbow cycle of length at least 4.

Heterochromatic paths in edge colored graphs without small cycles and heterochromatic-triangle-free graphs

Conditions for the existence of heterochromatic Hamiltonian paths and cycles in edge colored graphs are well investigated in literature. A related problem in this domain is to obtain good lower bounds for the length of a maximum heterochromatic path in an edge colored graph G. This problem is also well explored by now and the lower bounds are often specified as functions of the minimum color degree of G–the minimum number of distinct colors occurring at edges incident to any vertex of G–denoted by ϑ(G).Initially, it was conjectured that the lower bound for the length of a maximum heterochromatic path for an edge colored graph G would be ⌈2ϑ(G)3⌉. Chen and Li (2005) showed that the length of a maximum heterochromatic path in an edge colored graph G is at least ϑ(G)−1, if 1≤ϑ(G)≤7, and at least ⌈3ϑ(G)5⌉+1, if ϑ(G)≥8. They conjectured that the tight lower bound would be ϑ(G)−1 and demonstrated some examples which achieve this bound. An unpublished manuscript from the same authors (Chen, Li) reported to show that if ϑ(G)≥8, then G contains a heterochromatic path of length at least ⌈2ϑ(G)3⌉+1.In this paper, we give lower bounds for the length of a maximum heterochromatic path in edge colored graphs without small cycles. We show that if G has no four cycles, then it contains a heterochromatic path of length at least ϑ(G)−o(ϑ(G)) and if the girth of G is at least 4log2(ϑ(G))+2, then it contains a heterochromatic path of length at least ϑ(G)−2, which is only one less than the bound conjectured by Chen and Li (2005). Other special cases considered include lower bounds for the length of a maximum heterochromatic path in edge colored bipartite graphs and triangle-free graphs: for triangle-free graphs we obtain a lower bound of ⌊5ϑ(G)6⌋ and for bipartite graphs we obtain a lower bound of ⌈6ϑ(G)−37⌉.In this paper, it is also shown that if the coloring is such that G has no heterochromatic triangles, then G contains a heterochromatic path of length at least ⌊13ϑ(G)17⌋. This improves the previously known ⌈3ϑ(G)4⌉ bound obtained by Chen and Li (2011). We also give a relatively shorter and simpler proof showing that any edge colored graph G contains a heterochromatic path of length at least ⌈2ϑ(G)3⌉.

An Edge-Colored Version of Dirac's Theorem

Let $G$ be an edge-colored graph. The minimum color degree $\delta^c(G)$ of $G$ is the largest integer $k$ such that for every vertex $v$, there are at least $k$ distinct colors on edges incident to $v$. We say that $G$ is properly colored if no two adjacent edges have the same color. In this paper, we show that every edge-colored graph $G$ with $\delta^c(G) \ge 2|G|/3$ contains a properly colored $2$-factor. Furthermore, we show that for any $\varepsilon > 0 $ there exists an integer $n_0$ such that every edge-colored graph $G$ with $|G| = n \ge n_0$ and $\delta^c(G) \ge ( 2/3 + \varepsilon ) n $ contains a properly colored cycle of length $\ell$ for every $3 \le \ell \le n$. This result is best possible in the sense that the statement is false for $\delta^c(G) < 2n/3$.

Rainbow Matching in Edge-Colored Graphs

A rainbow subgraph of an edge-colored graph is a subgraph whose edges have distinct colors. The color degree of a vertex $v$ is the number of different colors on edges incident to $v$. Wang and Li conjectured that for $k\geq 4$, every edge-colored graph with minimum color degree at least $k$ contains a rainbow matching of size at least $\left\lceil k/2 \right\rceil$. We prove the slightly weaker statement that a rainbow matching of size at least $\left\lfloor k/2 \right\rfloor$ is guaranteed. We also give sufficient conditions for a rainbow matching of size at least $\left\lceil k/2 \right\rceil$ that fail to hold only for finitely many exceptions (for each odd $k$).