Heterochromatic paths in edge colored graphs without small cycles and heterochromatic-triangle-free graphs (original) (raw)

Cycles and paths in edge-colored graphs with given degrees

Journal of Graph Theory, 2009

We study sufficient degree conditions under various aspects guarantying the existence of properly edge-colored cycles and paths in edge-colored simple graphs, multigraphs and random graphs. In particular, we prove that an edge-colored multigraph of order n on at least three colors and with minimum colored degree greater than or equal to n+1 2 has properly edge-colored cycles of all possible lengths, including hamiltonian cycles. Also, longest properly edge-colored paths and hamiltonian paths between given vertices are considered.

Path chromatic numbers of graphs

Journal of Graph Theory, 1989

Let the finite, simple, undirected graph G = (V(G), E ( G ) ) be vertexcolored. Denote the distinct colors by 1,2, . . . , c . Let 6 be the set of all vertices colored i and let (V,) be the subgraph of G induced by V,. The k-path chromatic number of G, denoted by x(G; Pk), is the least number c of distinct colors with which V ( G ) can be colored such that each connected component of V, is a path of order at most k , 1 I i 5 c. We obtain upper bounds for x ( G ; P') and x ( G ; P, ) for regular, planar, and outerplanar graphs.

Properly coloured Hamiltonian paths in edge-coloured complete graphs

Discrete Applied Mathematics, 1998

We consider edge-coloured complete graphs. A path or cycle Q is called properly coloured (PC) if any two adjacent edges of Q differ in colour. Our note is inspired by the following conjecture by B. Bollobás and P. Erdős (1976) : if G is an edge-coloured complete graph on n vertices in which the maximum monochromatic degree of every vertex is less than n/2 , then G contains a PC Hamiltonian cycle. We prove that if an edge-coloured complete graph contains a PC 2-factor then it has a PC Hamiltonian path. R. Häggkvist (1996) announced that every edge-coloured complete graph satisfying Bollobás-Erdős condition contains a PC 2-factor. These two results imply that every edge-coloured complete graph satisfying Bollobás-Erdős condition has a PC Hamiltonian path.

On path-chromatically unique graphs

Discrete Mathematics, 1996

The least number of colors needed to color the vertices of a graph G such that the vertices in each color class induces a linear forest is called the path-chromatic number of G, denoted by Zoo (G). If all such colorings of the vertices of G induce the same partitioning of the vertices of G, we say that G is path-chromatically unique. We prove here that there exist infinitely many path-chromatically unique graphs with path-chromatic number t, for each t >/1. We also show that pathchromatically unique graphs of order n and path-chromatic number t >i 2 exist only for n 1> 4t-1.

On monochromatic paths in edge-coloured digraphs

Journal of Combinatorial Theory, Series B, 1982

DEDICATED TO RICHARD K. GUY ON THE OCCASION OF HIS RETIREMENT Let G be a directed graph whose edges are coloured with two colours. Call a set S of vertices of G independent if no two vertices of S are connected by a monochromatic directed path. We prove that if G contains no monochromatic infinite outward path, then there is an independent set S of vertices of G such that, for every vertex x not in S, there is a monochromatic directed path from x to a vertex of S. In the event that G is infinite, the proof uses Zorn's lemma. The last part of the paper is concerned with the case when G is a tournament. Proof. Let the two colours be red and blue. We first introduce some notation. For distinct vertices x, y of G, xjred y will mean that there is a

On paths, trails and closed trails in edge-colored graphs

In this paper we deal from an algorithmic perspective with different questions regarding properly edge-colored (or PEC) paths, trails and closed trails. Given a c-edge-colored graph G c , we show how to polynomially determine, if any, a PEC closed trail subgraph whose number of visits at each vertex is specified before hand. As a consequence, we solve a number of interesting related problems. For instance, given subset S of vertices in G c , we show how to maximize in polynomial time the number of S-restricted vertex (resp., edge) disjoint PEC paths (resp., trails) in G c with endpoints in S. Further, if G c contains no PEC closed trails, we show that the problem of finding a PEC s-t trail visiting a given subset of vertices can be solved in polynomial time and prove that it becomes NP-complete if we are restricted to graphs with no PEC cycles.

An Erdős-Gallai type theorem for vertex colored graphs

Graphs and Combinatorics, 2019

While investigating odd-cycle free hypergraphs, Győri and Lemons introduced a colored version of the classical theorem of Erdős and Gallai on P k-free graphs. They proved that any graph G with a proper vertex coloring and no path of length 2k + 1 with end vertices of different colors has at most 2kn edges. We show that Erdős and Gallai's original sharp upper bound of kn holds for their problem as well. We also introduce a version of this problem for trees and present a generalization of the Erdős-Sós conjecture.

On s-t paths and trails in edge-colored graphs

Electronic Notes in Discrete Mathematics, 2009

In this paper we deal from an algorithmic perspective with different questions regarding monochromatic and properly edge-colored s-t paths/trails on edge-colored graphs. Given a c-edge-colored graph G c without properly edge-colored closed trails, we present a polynomial time procedure for the determination of properly edgecolored s-t trails visiting all vertices of G c a prescribed number of times. As an immediate consequence, we polynomially solve the Hamiltonian path (resp., Eulerian trail) problem for this particular class of graphs. In addition, we prove that to check whether G c contains 2 properly edge-colored s-t paths/trails with length at most L > 0 is NP-complete in the strong sense. Finally, we prove that, if G c is a general c-edge-colored graph, to find 2 monochromatic vertex disjoint s-t paths with different colors is NP-complete.

Alternating Path of length L in C- edges Colored Graphs

A graph G: = (V, E) is called c- edge colored if and only if there exists a mapping A between the set E (G), the edge set in the graph G, and a set C of colors, such that |C| = c, and A maps a color c in C to every edge e∈ E(G)|. A path P in a c- edge colored graph is called an alternating path if two adjacent edges of P differ in color. We propose a dynamic algorithm, which resolves the following problem: 1. Find an alternating path of length L in a c- edge colored complete graph between two given vertices s, t ∈ V (G). 2. Besides we show that the proved results for c = 2 in [2] for the xy- path is also true for c > 2. Thus we generalize these results. We also deal with the problem of the (s, t) –Cycles, for 2- edge colored presented in [7], in a c- edge colored graph with c > 2. We propose another algorithm that finds in polynomial time the set of all paths and cycles of length of k, where k = log (n) and n is the order of the graph G. Further, a method to find in polynomial time the set of all paths and cycles of length of k within a c- edge colored graph, where log(n) < k ≤ n. We also consider the problem to determine the length of a shortest path between two given vertices in c- edge colored complete graph. In this paper, we therefore deal with the following question: 3. Under which conditions can it be possible to reduce in a graph G the length of a given path between two vertices such as the path remains alternating between the two vertices? In another paper we will deal with the minimal length of such path. The resolution of this question obliges us to consider the different classes of graphs to find the necessary and sufficient conditions to attain such goal.

Improved bounds on coloring of graphs

European Journal of Combinatorics, 2012

Given a graph G with maximum degree ∆, we prove that the acyclic edge chromatic number a ′ (G) of G is such that a ′ (G) ≤ 9.62∆. Moreover we prove that a ′ (G) ≤ 6.3∆ if G has girth g ≥ 5, a ′ (G) ≤ 5, 72∆ if G has girth g ≥ 7, a ′ (G) ≤ 4.52∆ if g ≥ 50 and a ′ (G) ≤ ∆ + 2 if g ≥ 26, 32∆ log ∆+ o(∆ log ∆). We further prove that the acyclic vertex chromatic number a(G) of G is such that a(G) ≤ 7, 213∆ 4/3 + o(∆ 4/3). We also prove that the star-chromatic number χ(G) of G is such that χ(G) ≤ √ 6∆ 3/2 + o(∆ 3/2). We finally prove that the β-frugal chromatic number χ β (G) of G is such that χ β (G) ≤ max{k 1 (β)∆, k 2 (β)∆ 1+1/β /(β!) 1/β }, where k 1 (β) and k 2 (β) are decreasing functions of β such that k 1 (β) ∈ [4, 6] and k 2 (β) ∈ [2, 5]. To obtain these results we use an improved version of the Lovász Local Lemma recently discovered.