Cycle extension in edge-colored complete graphs

Li, Ruonan; Broersma, Hajo; Xu, Chuandong; Zhang, Shenggui

doi:10.1016/j.disc.2017.01.023

Cited by 12 publications

(10 citation statements)

References 11 publications

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“…Recently, Lo [19] established the existence of a PC Hamilton cycle (a PC cycle of length n) in G for sufficiently large n when δ c (G) ≥ (1/2 + )n for any arbitrarily small constant > 0. In this context, in [15] the following cycle extension theorem was proved.…”

Section: Pc Theta Graphs Of Large Ordermentioning

confidence: 97%

“…Similarly, the existence of PC triangles has been studied by different research groups during the same period as well [6,8,[10][11][12]. These topics have been dealt with for general graphs [6,9,11,12,[16][17][18]23] but also for complete graphs [1,2,[4][5][6][7][8]10,14,15,19,21] and for complete bipartite graphs [1,4,15]. Moreover, the theory involved in the study of PC cycles in edge-colored graph is closely related to the theory of directed cycles in directed graphs.…”

Section: Introductionmentioning

confidence: 99%

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

Broersma

Zhang

2018

Graphs and Combinatorics

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With respect to specific cycle-related problems, edge-colored graphs can be considered as a generalization of directed graphs. We show that properly edge-colored theta graphs play a key role in characterizing the difference between edge-colored complete graphs and multipartite tournaments. We also establish sufficient conditions for an edgecolored complete graph to contain a small and a large properly edge-colored theta graph, respectively. Keywords Edge-colored graph • Complete graph • Properly edge-colored cycle • Properly edge-colored theta graph • Multipartite tournament Mathematics Subject Classification 05C15 • 05C20 • 05C38 Supported by CSC (No. 201506290097) and NSFC (No. 11671320).

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Section: Pc Theta Graphs Of Large Ordermentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

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

Broersma

Zhang

2018

Graphs and Combinatorics

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“…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

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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.

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“…Recently, Lo [47] established the existence of a PC Hamilton cycle (a PC cycle of length n) in G for sufficiently large n when δ c (G) ≥ (1/2 + ε)n for any arbitrarily small constant ε > 0. In this context, in [42] the following cycle extension theorem was proved.…”

Section: Claim 6 For Each Edge X Y ∈ E(g)mentioning

confidence: 99%

“…Similarly, the existence of PC triangles has been studied by different research groups during the same period as well [27,32,34,38,39]. These topics have been dealt with for general graphs [27,33,38,39,[44][45][46]65] but also for complete graphs [5,15,20,24,27,28,32,34,42,43,47,53] and for complete bipartite graphs [5,20,42]. Moreover, the theory involved in the study of PC cycles in edge-colored graph is closely related to the theory of directed cycles in directed graphs.…”

Cycle extension in edge-colored complete graphs

Cited by 12 publications

References 11 publications

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

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

Supereulerian 2-edge-coloured graphs

Properly colored cycles in edge-colored graphs

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