Properly Edge-colored Theta Graphs in Edge-colored Complete Graphs

Li, Ruonan; Broersma, Hajo; Zhang, Shenggui

doi:10.1007/s00373-018-1989-2

Cited by 15 publications

(8 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Generalized Theta graphs, which have been widely studied for many graph theoretic problems (see e.g., [8,11,14,22,23,27]), are the main subject of two classical papers on the chromatic polynomial [10] and [28] which include the celebrated result that the zeros of the chromatic polynomials of the Generalized Theta graphs are dense in the whole complex plane with the possible exception of the unit disc around the origin (by including the join of Generalized Theta graphs with K 2 this extends to all of the complex plane). It is natural to study the DP color function of these graphs, independent of our motivating questions.…”

Section: Summary Of Resultsmentioning

confidence: 99%

On Polynomial Representations of the DP Color Function: Theta Graphs and Their Generalizations

Halberg¹,

Kaul²,

Liu³

et al. 2020

Preprint

View full text Add to dashboard Cite

DP-coloring (also called correspondence coloring) is a generalization of list coloring that has been widely studied in recent years after its introduction by Dvořák and Postle in 2015. As the analogue of the chromatic polynomial P (G, m), the DP color function of a graph G, denoted P DP (G, m), counts the minimum number of DP-colorings over all possible m-fold covers. It is known that, unlike the list color function P ℓ (G, m), for any g ≥ 3 there exists a graph G with girth g such that P DP (G, m) < P (G, m) when m is sufficiently large. Thus, two fundamental open questions regarding the DP color function are: (i) for which G does there exist an N ∈ N such that P DP (G, m) = P (G, m) whenever m ≥ N , (ii) Given a graph G does there always exist an N ∈ N and a polynomial p(m) such that P DP (G, m) = p(m) whenever m ≥ N ?In this paper we give exact formulas for the DP color function of a Theta graph based on the parity of its path lengths. This gives an explicit answer, including the formulas for the polynomials that are not the chromatic polynomial, to both the questions above for Theta graphs. We extend this result to Generalized Theta graphs by characterizing the exact parity condition that ensures the DP color function eventually equals the chromatic polynomial. To answer the second question for Generalized Theta graphs, we confirm it for the larger class of graphs with a feedback vertex set of size one.

show abstract

Section: Summary Of Resultsmentioning

confidence: 99%

On Polynomial Representations of the DP Color Function: Theta Graphs and Their Generalizations

Halberg¹,

Kaul²,

Liu³

et al. 2020

Preprint

View full text Add to dashboard Cite

show abstract

“…Although Li et al [13] and later Xu et al [21] have got pr(K n , C 4 ) = n for n ≥ 4, here we will use a different method to prove it again. We denote a cycle C k with a pendant edge by C + k .…”

Section: Cyclesmentioning

confidence: 92%

“…It is easy to see that pr(K n , C 3 ) = ar(K n , C 3 ) = n − 1. Li, Broersma and Zhang [13], and later Xu, Magnant, and Zhang [21] showed that pr(K n , C 4 ) = n for n ≥ 4. We consider C 5 and C 6 in section 4.…”

mentioning

confidence: 96%

Minimal colorings for properly colored subgraphs in complete graphs

Fang

Győri

Xiao

2019

Preprint

View full text Add to dashboard Cite

Let pr(K n , G) be the maximum number of colors in an edge-coloring of K n with no properly colored copy of G. In this paper, we show that pr, where G ′ = {G − M : M is a matching of G}. Furthermore, we determine the value of pr(K n , P l ) for l ≥ 27 and n ≥ 2l 3 and the exact value of pr(K n , G), where G is C 5 , C 6 and K − 4 , respectively. Also, we give an upper bound and a lower bound of pr(K n , K 2,3 ).

show abstract

“…in polynomial time. This has been the topic of many papers in the past, see e.g [4,7,8,9,10,11,12,17,18,19,22].…”

Section: Introductionmentioning

confidence: 99%

Supereulerian 2-edge-coloured graphs

Bang‐Jensen

Bellitto

Yeo

2020

Preprint

View full text Add to dashboard Cite

A 2-edge-coloured graph G is supereulerian if G contains a spanning closed trail in which the edges alternate in colours. An eulerian factor of a 2-edge-coloured graph is a collection of vertex disjoint induced subgraphs which cover all the vertices of G such that each of these subgraphs is supereulerian. We give a polynomial algorithm to test if a 2-edge-coloured graph has an eulerian factor and to produce one when it exists. A 2edge-coloured graph is (trail-)colour-connected if it contains a pair of alternating (u, v)-paths ((u, v)-trails) whose union is an alternating closed walk for every pair of distinct vertices u, v. A 2-edge-coloured graph is M-closed if xz is an edge of G whenever some vertex u is joined to both x and z by edges of the same colour. M-closed 2-edge-coloured graphs, introduced in [11], form a rich generalization of 2-edge-coloured complete graphs. We show that if G is an extension of an M-closed 2-edge-coloured complete graph, then G is supereulerian if and only if G is trail-colourconnected and has an eulerian factor. We also show that for general 2-edge-coloured graphs it is NP-complete to decide whether the graph is supereulerian. Finally we pose a number of open problems. Keywords: 2-edge-coloured graph; alternating hamiltonian cycle; supereulerian; alternating cycle; eulerian factor; extension of a 2-edge-coloured graph.

show abstract

Properly Edge-colored Theta Graphs in Edge-colored Complete Graphs

Cited by 15 publications

References 23 publications

On Polynomial Representations of the DP Color Function: Theta Graphs and Their Generalizations

On Polynomial Representations of the DP Color Function: Theta Graphs and Their Generalizations

Minimal colorings for properly colored subgraphs in complete graphs

Supereulerian 2-edge-coloured graphs

Contact Info

Product

Resources

About