11 research outputs found
Acyclic 4-choosability of planar graphs without 4-cycles
summary:A proper vertex coloring of a graph is acyclic if there is no bicolored cycle in . In other words, each cycle of must be colored with at least three colors. Given a list assignment , if there exists an acyclic coloring of such that for all , then we say that is acyclically -colorable. If is acyclically -colorable for any list assignment with for all , then is acyclically -choosable. In 2006, Montassier, Raspaud and Wang conjectured that every planar graph without 4-cycles is acyclically 4-choosable. However, this has been as yet verified only for some restricted classes of planar graphs. In this paper, we prove that every planar graph with neither 4-cycles nor intersecting -cycles for each is acyclically 4-choosable
Acyclic 5-Choosability of Planar Graphs Without Adjacent Short Cycles
The conjecture claiming that every planar graph is acyclic 5-choosable[Borodin et al., 2002] has been verified for several restricted classes of planargraphs. Recently, O. V. Borodin and A. O. Ivanova, [Journal of Graph Theory,68(2), October 2011, 169-176], have shown that a planar graph is acyclically 5-choosable if it does not contain an i-cycle adjacent to a j-cycle, where 3<=j<=5 if i=3 and 4<=j<=6 if i=4. We improve the above mentioned result and prove that every planar graph without an i-cycle adjacent to a j-cycle with3<=j<=5 if i=3 and 4<=j<=5 if i=4 is acyclically 5-choosable
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Problems of optimal choice on posets and generalizations of acyclic colourings
NOTE : The mathematical symbols in the abstract do not always display correctly in this text field. Please see the abstract in the thesis for the definitive abstract. ABSTRACT: This dissertation is in two parts, each of three chapters. In Part 1, I shall prove some results concerning variants of the `secretary problem'. In Part 2, I shall bound several generalizations of the acyclic chromatic number of a graph as functions of its maximum degree.
I shall begin Chapter 1 by describing the classical secretary problem, in which the aim is to select the best candidate for the post of a secretary, and its solution. I shall then summarize some of its many generalizations that have been studied up to now, provide some basic theory, and briefly outline the results that I shall prove.
In Chapter 2, I shall suppose that the candidates come as âmâ pairs of equally qualified identical twins. I shall describe an optimal strategy, a formula for its probability of success and the asymptotic behaviour of this strategy and its probability of success as m â â. I shall also find an optimal strategy and its probability of success for the analagous version with âcâ-tuplets.
I shall move away from known posets in Chapter 3, assuming instead that the candidates come from a poset about which the only information known is its size and number of maximal elements. I shall show that, given this information, there is an algorithm that is successful with probability at least Âč/e . For posets with âk â„ 2â maximal elements, I shall prove that if their width is also âkâ then this can be improved to âk-1â1/kâ and show that no better bound of this type is possible.
In Chapter 4, I shall describe the history of acyclic colourings, in which a graph must be properly coloured with no two-coloured cycle, and state some results known about them and their variants. In particular, I shall highlight a result of Alon, McDiarmid and Reed, which bounds the acyclic chromatic number of a graph by a function of its maximum degree. My results in the next two chapters are of this form.
I shall consider two natural generalizations in Chapter 5. In the first, only cycles of length at least âlâ must receive at least three colours. In the second, every cycle must receive at least âcâ colours, except those of length less than âcâ, which must be multicoloured.
My results in Chapter 6 generalize the concept of a cycle; it is now subgraphs with minimum degree ârâ that must receive at least three colours, rather than subgraphs with minimum degree two (which contain cycles). I shall also consider a natural version of this problem for hypergraphs
Graph coloring under constraints
Dans cette thĂšse, nous nous intĂ©ressons Ă diffĂ©rentes notions de colorations sous contraintes. Nous nous intĂ©ressons plus spĂ©cialement Ă la coloration acyclique, Ă la coloration forte d'arĂȘtes et Ă la coloration d'arĂȘtes sommets adjacents distinguants.Dans le Chapitre 2, nous avons Ă©tudiĂ© la coloration acyclique. Tout d'abord nous avons cherchĂ© Ă borner le nombre chromatique acyclique pour la classe des graphes de degrĂ© maximum bornĂ©. Ensuite nous nous sommes attardĂ©s sur la coloration acyclique par listes. La notion de coloration acyclique par liste des graphes planaires a Ă©tĂ© introduite par Borodin, Fon-Der Flaass, Kostochka, Raspaud et Sopena. Ils ont conjecturĂ© que tout graphe planaire est acycliquement 5-liste coloriable. De notre cĂŽtĂ©, nous avons proposĂ© des conditions suffisantes de 3-liste coloration acyclique des graphes planaires. Dans le Chapitre 3, nous avons Ă©tudiĂ© la coloration forte d'arĂȘtes des graphes subcubiques en majorant l'indice chromatique fort en fonction du degrĂ© moyen maximum. Nous nous sommes Ă©galement intĂ©ressĂ©s Ă la coloration forte d'arĂȘtes des graphes subcubiques sans cycles de longueurs donnĂ©es et nous avons Ă©galement obtenu une majoration optimale de l'indice chromatique fort pour la famille des graphes planaires extĂ©rieurs. Nous avons aussi prĂ©sentĂ© diffĂ©rents rĂ©sultats de complexitĂ© pour la classe des graphes planaires subcubiques. Enfin, au Chapitre 4, nous avons abordĂ© la coloration d'arĂȘtes sommets adjacents distinguants en dĂ©terminant les majorations de l'indice avd-chromatique en fonction du degrĂ© moyen maximum. Notre travail s'inscrit dans la continuitĂ© de celui effectuĂ© par Wang et Wang en 2010. Plus prĂ©cisĂ©ment, nous nous sommes focalisĂ©s sur la famille des graphes de degrĂ© maximum au moins 5.In this thesis, we are interested in various coloring of graphs under constraints. We study acyclic coloring, strong edge coloring and adjacent vertex-distinguishing edge coloring.In Chapter 2, we consider acyclic coloring and we bound the acyclic chromatic number by a function of the maximum degree of the graph. We also study acyclic list coloring. The notion of acyclic list coloring of planar graphs was introduced by Borodin, Fon-Der Flaass, Kostochka, Raspaud, and Sopena. They conjectured that every planar graph is acyclically 5-choosable. We obtain some sufficient conditions for planar graphs to be acyclically 3-choosable.In Chapter 3, we study strong edge coloring of graphs. We prove some upper bounds of the strong chromatic index of subcubic graphs as a function of the maximum average degree. We also obtain a tight upper bound for the minimum number of colors in a strong edge coloring of outerplanar graphs as a function of the maximum degree. We also prove that the strong edge k-colouring problem, when k=4,5,6, is NP-complete for subcubic planar bipartite graphs with some girth condition. Finally, in Chapter 4, we focus on adjacent vertex-distinguishing edge coloring, or avd-coloring, of graphs. We bound the avd-chromatic number of graphs by a function of the maximum average degree. This work completes a result of Wang and Wang in 2010
Graph coloring under constraints
Dans cette thĂšse, nous nous intĂ©ressons Ă diffĂ©rentes notions de colorations sous contraintes. Nous nous intĂ©ressons plus spĂ©cialement Ă la coloration acyclique, Ă la coloration forte d'arĂȘtes et Ă la coloration d'arĂȘtes sommets adjacents distinguants.Dans le Chapitre 2, nous avons Ă©tudiĂ© la coloration acyclique. Tout d'abord nous avons cherchĂ© Ă borner le nombre chromatique acyclique pour la classe des graphes de degrĂ© maximum bornĂ©. Ensuite nous nous sommes attardĂ©s sur la coloration acyclique par listes. La notion de coloration acyclique par liste des graphes planaires a Ă©tĂ© introduite par Borodin, Fon-Der Flaass, Kostochka, Raspaud et Sopena. Ils ont conjecturĂ© que tout graphe planaire est acycliquement 5-liste coloriable. De notre cĂŽtĂ©, nous avons proposĂ© des conditions suffisantes de 3-liste coloration acyclique des graphes planaires. Dans le Chapitre 3, nous avons Ă©tudiĂ© la coloration forte d'arĂȘtes des graphes subcubiques en majorant l'indice chromatique fort en fonction du degrĂ© moyen maximum. Nous nous sommes Ă©galement intĂ©ressĂ©s Ă la coloration forte d'arĂȘtes des graphes subcubiques sans cycles de longueurs donnĂ©es et nous avons Ă©galement obtenu une majoration optimale de l'indice chromatique fort pour la famille des graphes planaires extĂ©rieurs. Nous avons aussi prĂ©sentĂ© diffĂ©rents rĂ©sultats de complexitĂ© pour la classe des graphes planaires subcubiques. Enfin, au Chapitre 4, nous avons abordĂ© la coloration d'arĂȘtes sommets adjacents distinguants en dĂ©terminant les majorations de l'indice avd-chromatique en fonction du degrĂ© moyen maximum. Notre travail s'inscrit dans la continuitĂ© de celui effectuĂ© par Wang et Wang en 2010. Plus prĂ©cisĂ©ment, nous nous sommes focalisĂ©s sur la famille des graphes de degrĂ© maximum au moins 5.In this thesis, we are interested in various coloring of graphs under constraints. We study acyclic coloring, strong edge coloring and adjacent vertex-distinguishing edge coloring.In Chapter 2, we consider acyclic coloring and we bound the acyclic chromatic number by a function of the maximum degree of the graph. We also study acyclic list coloring. The notion of acyclic list coloring of planar graphs was introduced by Borodin, Fon-Der Flaass, Kostochka, Raspaud, and Sopena. They conjectured that every planar graph is acyclically 5-choosable. We obtain some sufficient conditions for planar graphs to be acyclically 3-choosable.In Chapter 3, we study strong edge coloring of graphs. We prove some upper bounds of the strong chromatic index of subcubic graphs as a function of the maximum average degree. We also obtain a tight upper bound for the minimum number of colors in a strong edge coloring of outerplanar graphs as a function of the maximum degree. We also prove that the strong edge k-colouring problem, when k=4,5,6, is NP-complete for subcubic planar bipartite graphs with some girth condition. Finally, in Chapter 4, we focus on adjacent vertex-distinguishing edge coloring, or avd-coloring, of graphs. We bound the avd-chromatic number of graphs by a function of the maximum average degree. This work completes a result of Wang and Wang in 2010
Colorations de graphes sous contraintes
Dans cette thĂšse, nous nous intĂ©ressons Ă diffĂ©rentes notions de colorations sous contraintes. Nous nous intĂ©ressons plus spĂ©cialement Ă la coloration acyclique, Ă la coloration forte d'arĂȘtes et Ă la coloration d'arĂȘtes sommets adjacents distinguants.Dans le Chapitre 2, nous avons Ă©tudiĂ© la coloration acyclique. Tout d'abord nous avons cherchĂ© Ă borner le nombre chromatique acyclique pour la classe des graphes de degrĂ© maximum bornĂ©. Ensuite nous nous sommes attardĂ©s sur la coloration acyclique par listes. La notion de coloration acyclique par liste des graphes planaires a Ă©tĂ© introduite par Borodin, Fon-Der Flaass, Kostochka, Raspaud et Sopena. Ils ont conjecturĂ© que tout graphe planaire est acycliquement 5-liste coloriable. De notre cĂŽtĂ©, nous avons proposĂ© des conditions suffisantes de 3-liste coloration acyclique des graphes planaires. Dans le Chapitre 3, nous avons Ă©tudiĂ© la coloration forte d'arĂȘtes des graphes subcubiques en majorant l'indice chromatique fort en fonction du degrĂ© moyen maximum. Nous nous sommes Ă©galement intĂ©ressĂ©s Ă la coloration forte d'arĂȘtes des graphes subcubiques sans cycles de longueurs donnĂ©es et nous avons Ă©galement obtenu une majoration optimale de l'indice chromatique fort pour la famille des graphes planaires extĂ©rieurs. Nous avons aussi prĂ©sentĂ© diffĂ©rents rĂ©sultats de complexitĂ© pour la classe des graphes planaires subcubiques. Enfin, au Chapitre 4, nous avons abordĂ© la coloration d'arĂȘtes sommets adjacents distinguants en dĂ©terminant les majorations de l'indice avd-chromatique en fonction du degrĂ© moyen maximum. Notre travail s'inscrit dans la continuitĂ© de celui effectuĂ© par Wang et Wang en 2010. Plus prĂ©cisĂ©ment, nous nous sommes focalisĂ©s sur la famille des graphes de degrĂ© maximum au moins 5.In this thesis, we are interested in various coloring of graphs under constraints. We study acyclic coloring, strong edge coloring and adjacent vertex-distinguishing edge coloring.In Chapter 2, we consider acyclic coloring and we bound the acyclic chromatic number by a function of the maximum degree of the graph. We also study acyclic list coloring. The notion of acyclic list coloring of planar graphs was introduced by Borodin, Fon-Der Flaass, Kostochka, Raspaud, and Sopena. They conjectured that every planar graph is acyclically 5-choosable. We obtain some sufficient conditions for planar graphs to be acyclically 3-choosable.In Chapter 3, we study strong edge coloring of graphs. We prove some upper bounds of the strong chromatic index of subcubic graphs as a function of the maximum average degree. We also obtain a tight upper bound for the minimum number of colors in a strong edge coloring of outerplanar graphs as a function of the maximum degree. We also prove that the strong edge k-colouring problem, when k=4,5,6, is NP-complete for subcubic planar bipartite graphs with some girth condition. Finally, in Chapter 4, we focus on adjacent vertex-distinguishing edge coloring, or avd-coloring, of graphs. We bound the avd-chromatic number of graphs by a function of the maximum average degree. This work completes a result of Wang and Wang in 2010.BORDEAUX1-Bib.electronique (335229901) / SudocSudocFranceF