On the structure and deficiency of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si52.gif" display="inline" overflow="scroll"><mml:mi>k</mml:mi></mml:math>-trees with bounded degree

Borowiecka-Olszewska, Marta; Drgas-Burchardt, Ewa; Hałuszczak, Mariusz

doi:10.1016/j.dam.2015.08.008

Cited by 8 publications

(8 citation statements)

References 14 publications

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“…In general, the lower bound on def (K 2n+1 −mK 2 ) (1 ≤ m ≤ n) is not sharp, since for any maximum matching M of K 2n+1 (n ≥ 2), K 2n+1 −M / ∈ N [25], hence def (K 2n+1 −nK 2 ) > 0. However, the lower bound is sharp for K 2n+1 − e as it was conjectured in [7]. Proof.…”

Section: The Deficiency Of Certain Graphsmentioning

confidence: 63%

“…Bouchard, Hertz and Desaulniers [9] derived some lower bounds for the deficiency of graphs and provided a tabu search algorithm for finding a proper edge-coloring with minimum deficiency of a graph. Recently, Borowiecka-Olszewska, Drgas-Burchardt and Ha luszczak [7] studied the deficiency of k-trees. In particular, they determined the deficiency of all k-trees with maximum degree at most 2k, where k ∈ {2, 3, 4}.…”

mentioning

confidence: 99%

“…[7] If G is a graph with an odd number of vertices, thendef (G) ≥ 2|E(G)|−(|V (G)|−1)∆(G)Corollary 4.2. [7] For any n, m ∈ N, we have…”

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confidence: 99%

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Further results on the deficiency of graphs

Khachatrian

2017

Discrete Applied Mathematics

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Section: The Deficiency Of Certain Graphsmentioning

confidence: 63%

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confidence: 99%

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Further results on the deficiency of graphs

Khachatrian

2017

Discrete Applied Mathematics

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“…Schwartz [33] obtained tight bounds on the deficiency of some families of regular graphs. Recently, Petrosyan and Khachatrian [31] proved that for near-complete graphs def(K 2n+1 − e) = n − 1 (where e is an edge of K 2n+1 ), thereby confirming a conjecture of Borowiecka-Olszewska et al [7]. Further results on deficiency appear in [1,7,8,9,13,26,31].…”

Section: Introductionmentioning

confidence: 74%

Cyclic deficiency of graphs

Asratian

Casselgren

2019

Discrete Applied Mathematics

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A proper edge coloring of a graph G with colors 1, 2, . . . , t is called a cyclic interval t-coloring if for each vertex v of G the edges incident to v are colored by consecutive colors, under the condition that color 1 is considered as consecutive to color t. In this paper we introduce and investigate a new notion, the cyclic deficiency of a graph G, defined as the minimum number of pendant edges whose attachment to G yields a graph admitting a cyclic interval coloring; this number can be considered as a measure of closeness of G of being cyclically interval colorable. We determine or bound the cyclic deficiency of several families of graphs. In particular, we present examples of graphs of bounded maximum degree with arbitrarily large cyclic deficiency, and graphs whose cyclic deficiency approaches the number of vertices. Finally, we conjecture that the cyclic deficiency of any graph does not exceed the number of vertices, and we present several results supporting this conjecture.

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“…Also some classes of biregular graphs (see e.g. [9,10,16,17,20]), k-trees [8], generalized Hertz graphs [6], and generalized Sevastjanov rosettes [7] have consecutive colourings. There are many papers devoted to this topic, in particular surveys which can be found in the books [1,22].…”

Section: Motivationmentioning

confidence: 99%

Consecutive Colouring of Oriented Graphs

et al. 2021

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We consider arc colourings of oriented graphs such that for each vertex the colours of all out-arcs incident with the vertex and the colours of all in-arcs incident with the vertex form intervals. We prove that the existence of such a colouring is an NP-complete problem. We give the solution of the problem for r-regular oriented graphs, transitive tournaments, oriented graphs with small maximum degree, oriented graphs with small order and some other classes of oriented graphs. We state the conjecture that for each graph there exists a consecutive colourable orientation and confirm the conjecture for complete graphs, 2-degenerate graphs, planar graphs with girth at least 8, and bipartite graphs with arboricity at most two that include all planar bipartite graphs. Additionally, we prove that the conjecture is true for all perfect consecutively colourable graphs and for all forbidden graphs for the class of perfect consecutively colourable graphs.

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On the structure and deficiency of k-trees with bounded degree

Cited by 8 publications

References 14 publications

Further results on the deficiency of graphs

Further results on the deficiency of graphs

Cyclic deficiency of graphs

Consecutive Colouring of Oriented Graphs

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