Odd properly colored cycles in edge-colored graphs

Gutin, Gregory; Sheng, Bin; Wahlström, Magnus

doi:10.1016/j.disc.2016.11.017

Cited by 6 publications

(3 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Xu et al [10] determined the structure of an

n

‐colored

K_{n}

containing no PC

C_{4}

and gave sufficient conditions for the existence of PC

C_{4}

's in edge‐colored graphs. From a computational complexity angle, Gutin et al [7] studied the complexity of determining the existence of odd PC cycles in edge‐colored graphs.…”

Section: Introductionmentioning

confidence: 99%

Edge‐colored complete graphs without properly colored even cycles: A full characterization

Broersma

Yokota

et al. 2021

Journal of Graph Theory

View full text Add to dashboard Cite

The structure of edge-colored complete graphs containing no properly colored triangles has been characterized by Gallai back in the 1960s. More recently, Cǎda et al. and Fujita et al. independently determined the structure of edge-colored complete bipartite graphs containing no properly colored C 4 . We characterize the structure of edge-colored complete graphs containing no properly colored even cycles, or equivalently, without a properly colored C 4 or C 6 . In particular, we first deal with the simple case of 2-edge-colored complete graphs, using a result of Yeo. Next, for ≥ k 3, we define four classes of k-edgecolored complete graphs without properly colored even cycles and prove that any k-edge-colored complete graph without a properly colored even cycle belongs to one of these four classes.This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made.

show abstract

“…Xu et al [10] determined the structure of an

n

‐colored

K_{n}

containing no PC

C_{4}

and gave sufficient conditions for the existence of PC

C_{4}

's in edge‐colored graphs. From a computational complexity angle, Gutin et al [7] studied the complexity of determining the existence of odd PC cycles in edge‐colored graphs.…”

Section: Introductionmentioning

confidence: 99%

Edge‐colored complete graphs without properly colored even cycles: A full characterization

Broersma

Yokota

et al. 2021

Journal of Graph Theory

View full text Add to dashboard Cite

show abstract

“…There are many positive algorithmic results on PC walks in edge-colored graphs, for a detailed survey, we refer interested readers to Chapter 16 of [4] for pre-2009 literature, and to, e.g., [1,10,16,19,22,23] for later publications. Unfortunately, most problems turn out to be much harder for edge-colored digraphs than for edge-colored undirected graphs.…”

Section: Introductionmentioning

confidence: 99%

The Euler and Chinese Postman Problems on 2-Arc-Colored Digraphs

Sheng,

Li,

Gutin

2017

Preprint

Self Cite

View full text Add to dashboard Cite

The famous Chinese Postman Problem (CPP) is polynomial time solvable on both undirected and directed graphs. Gutin et al. [Discrete Applied Math 217 (2016)] generalized these results by proving that CPP on c-edge-colored graphs is polynomial time solvable for every c ≥ 2. In CPP on weighted edge-colored graphs G, we wish to find a minimum weight properly colored closed walk containing all edges of G (a walk is properly colored if every two consecutive edges are of different color, including the last and first edges in a closed walk). In this paper, we consider CPP on arc-colored digraphs (for properly colored closed directed walks), and provide a polynomial-time algorithm for the problem on weighted 2-arc-colored digraphs. This is a somewhat surprising result since it is NP-complete to decide whether a 2-arc-colored digraph has a properly colored directed cycle [Gutin et al., Discrete Math 191 (1998)]. To obtain the polynomial-time algorithm, we characterize 2-arc-colored digraphs containing properly colored Euler trails.

show abstract

“…According to this characterization, we obtain the following immediate corollary. Gutin et al [51] in 2017 studied the problem of deciding the existence of a PC odd cycle in an edge-colored graph. Until now, the existence of a deterministic polynomial time algorithm for a PC odd cycle in an edge-colored graph is still an open question.…”

Section: Introductionmentioning

confidence: 99%

Properly colored subgraphs in edge-colored graphs

Han

View full text Add to dashboard Cite

show abstract

Odd properly colored cycles in edge-colored graphs

Cited by 6 publications

References 19 publications

Edge‐colored complete graphs without properly colored even cycles: A full characterization

Edge‐colored complete graphs without properly colored even cycles: A full characterization

The Euler and Chinese Postman Problems on 2-Arc-Colored Digraphs

Properly colored subgraphs in edge-colored graphs

Contact Info

Product

Resources

About