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Equitable Colourings of d-degenerate Graphs

A. V. KOSTOCHKA, K. NAKPRASIT
2003 Combinatorics, probability & computing  
A proper vertex coloring of a graph is called equitable if the sizes of colour classes differ by at most 1.  ...  In this paper, we find the minimum number l = l(d, ∆) such that every d -degenerate graph with maximum degree at most ∆ admits an equitable t-colouring for every t l when ∆ 27d.  ...  We will show that ν(d, ∆) colours is enough for an equitable colouring of an arbitrary d -degenerate graph with maximum degree ∆ provided that ∆/d is large. Theorem 1.1.  ... 
doi:10.1017/s0963548302005485 fatcat:fwiz4krpwzcthpupn573hylxwi

Equitable d-degenerate Choosability of Graphs [chapter]

Ewa Drgas-Burchardt, Hanna Furmańczyk, Elżbieta Sidorowicz
2020 Lecture Notes in Computer Science  
Let D d be the class of d-degenerate graphs and let L be a list assignment for a graph G.  ...  A colouring of G such that every vertex receives a colour from its list and the subgraph induced by vertices coloured with one color is a d-degenerate graph is called the (L, D d )colouring of G.  ...  For d ∈ N 0 , the graph G is d-degenerate if δ(H) ≤ d for any subgraph H of G, where δ(H) denotes the minimum degree of H. The class of all d-degenerate graphs is denoted by D d .  ... 
doi:10.1007/978-3-030-48966-3_19 fatcat:uo5s5a24zrcmpegxcdyltuyukm

Equitable d-degenerate choosability of graphs [article]

E. Drgas-Burchardt, H. Furmańczyk, E. Sidorowicz
2020 arXiv   pre-print
Let D_d be the class of d-degenerate graphs and let L be a list assignment for a graph G.  ...  A colouring of G such that every vertex receives a colour from its list and the subgraph induced by vertices coloured with one color is a d-degenerate graph is called the (L, D_d)-colouring of G.  ...  For d ∈ N 0 , the graph G is d-degenerate if δ(H) ≤ d for any subgraph H of G, where δ(H) denotes the minimum degree of H. The class of all d-degenerate graphs is denoted by D d .  ... 
arXiv:2003.09722v1 fatcat:4y4jbnhbmra3hepky7b44jod7m

Equitable list vertex colourability and arboricity of grids

Ewa Drgas-Burchardt, Janusz Dybizbański, Hanna Furmańczyk, Elżbieta Sidorowicz
2018 Filomat  
A graph G is equitably k-list arborable if for any k-uniform list assignment L, there is an equitable L-colouring of G whose each colour class induces an acyclic graph.  ...  We give some new tools that are helpful in determining values of k for which a general graph is equitably k-list arborable.  ...  There is a graph G and k ∈ N such that G is k-list arborable and G has a k-colouring in which each colour class is of the cardinality at most |V(G)|/k and induces an acyclic graph, however G is not equitably  ... 
doi:10.2298/fil1818353d fatcat:2wc26mgl3rgn3moxkdnphwoqc4

Equitable List Vertex Colourability and Arboricity of Grids [article]

Ewa Drgas-Burchardt, Janusz Dybizbański, Hanna Furmańczyk, Elzbieta Sidorowicz
2018 arXiv   pre-print
A graph G is equitably k-list arborable if for any k-uniform list assignment L, there is an equitable L-colouring of G whose each colour class induces an acyclic graph.  ...  We give some new tools that are helpful in determining values of k for which a general graph is equitably k-list arborable.  ...  Among others, d-dimensional grids are bipartite and d-degenerate. The equitable colouring of such classes of graphs is analyzed in many papers.  ... 
arXiv:1809.08281v1 fatcat:auwso2nbofh6hnoaz7du53nvfq

Page 2115 of Mathematical Reviews Vol. 57, Issue 6 [page]

1979 Mathematical Reviews  
A theorem, relating the chromatic number of a graph and of one of its bipartition kernels, is given. An algorithm for graph colouring, based on this theorem, is proposed. D.  ...  A graph G is said to be n- degenerate if each subgroup of G_has minimum degree at most n.  ... 

Independent Coverings and Orthogonal Colourings [article]

Kyle MacKeigan
2021 arXiv   pre-print
One conjecture regards independent coverings of sparse partite graphs, whereas the other conjecture regards orthogonal colourings of tree graphs.  ...  A relation between independent coverings and orthogonal colourings is established. This relation is applied to find independent coverings of some sparse partite graphs.  ...  Such an ordering of the vertices is called a degenerate ordering. The degeneracy of a graph is the smallest d for which the graph is d-degenerate.  ... 
arXiv:2008.07904v3 fatcat:seagrwz7fvgqrbc3ba4af3o3va

Equitable colorings of $K_4$-minor-free graphs

Rémi de Joannis de Verclos, Jean-Sébastien Sereni
2017 Journal of Graph Algorithms and Applications  
We do not use the discharging method but rather exploit decomposition trees of K 4 -minor-free graphs.  ...  We demonstrate that for every positive integer ∆, every K 4 -minor-free graph with maximum degree ∆ admits an equitable coloring with k colors where k ∆+3 2 .  ...  This contradicts our assumption that |V (G)| > k, thereby finishing the proof of Theorem 1.4.  ... 
doi:10.7155/jgaa.00451 fatcat:kp2bugfz7ba77mmgu2ym7numgm

Equitable Colorings of K_4-minor-free Graphs [article]

Rémi De Joannis de Verclos
2017 arXiv   pre-print
We do not use the discharging method but rather exploit decomposition trees of K 4-minor-free graphs.  ...  We demonstrate that for every positive integer Δ, every K_4-minor-free graph with maximum degree Δ admits an equitable coloring with k colors wherek > (Δ+3)/2.  ...  This contradicts our assumption that |V (G)| > k, thereby finishing the proof of Theorem 1.4.  ... 
arXiv:1703.02250v1 fatcat:pjlsrohssnf2zas4x5cduxthoq

Spanning embeddings of arrangeable graphs with sublinear bandwidth [article]

Julia Böttcher, Anusch Taraz, Andreas Würfl
2013 arXiv   pre-print
We generalise this result to a-arrangeable graphs H with \Delta(H)<sqrt(n)/log(n), where n is the number of vertices of H.  ...  The Bandwidth Theorem of B\"ottcher, Schacht and Taraz [Mathematische Annalen 343 (1), 175-205] gives minimum degree conditions for the containment of spanning graphs H with small bandwidth and bounded  ...  Indeed, a version of the Blow-up Lemma for d-degenerate graphs would imply the difficult and long-standing Burr-Erdös conjecture [6] , which states that degenerate graphs have linear Ramsey number.  ... 
arXiv:1305.2078v1 fatcat:fxskstnyffcw3cymldtgddmij4

Spanning embeddings of arrangeable graphs with sublinear bandwidth

Julia Böttcher, Anusch Taraz, Andreas Würfl
2015 Random structures & algorithms (Print)  
We generalise this result to a-arrangeable graphs H with ∆(H) ≤ √ n/ log n, where n is the number of vertices of H.  ...  The Bandwidth Theorem of Böttcher, Schacht, and Taraz [Mathematische Annalen 343 (1), gives minimum degree conditions for the containment of spanning graphs H with small bandwidth and bounded maximum degree  ...  Moreover, we are grateful for the careful reading and valuable comments of two anonymous referees.  ... 
doi:10.1002/rsa.20593 fatcat:doc423tz7neqfjbiorduj44uey

The Bandwidth Theorem in Sparse Graphs [article]

Peter Allen, Julia Böttcher, Julia Ehrenmüller, Anusch Taraz
2020 arXiv   pre-print
More precisely, we show that for p≫(log nn)^1/Δ asymptotically almost surely each spanning subgraph G of G(n,p) with minimum degree (k-1k+o(1))pn contains all n-vertex k-colourable graphs H with maximum  ...  We provide sparse analogues of this statement in random graphs as well as pseudorandom graphs.  ...  Let H be a D-degenerate graph on n vertices, let L be a labelling of its vertex set of bandwidth at most β n and let σ : V (H) → {0, . . . k} be a proper (k + 1)-colouring that is (10/ξ , β )zero-free  ... 
arXiv:1612.00661v2 fatcat:5etgg2uiqrapped3aqdegi4ufy

Towards Lehel's conjecture for 4-uniform tight cycles [article]

Allan Lo, Vincent Pfenninger
2020 arXiv   pre-print
A k-uniform tight cycle is a k-uniform hypergraph with a cyclic ordering of its vertices such that its edges are all the sets of size k formed by k consecutive vertices in the ordering.  ...  We prove that every red-blue edge-coloured K_n^(4) contains a red and a blue tight cycle that are vertex-disjoint and together cover n-o(n) vertices.  ...  Given a (t 0 , t 1 , ε)-equitable (k − 1)-complex J and a k-graph G on V (J ), we say that G is (ε k , r)-regular with respect to a k-set X of clusters of J if there exists some d such that G is (d, ε  ... 
arXiv:2012.08875v1 fatcat:qgrk526snbhonfrburywvollpy

Threshold functions for asymmetric Ramsey properties involving cycles

Y. Kohayakawa, B. Kreuter
1997 Random structures & algorithms (Print)  
As deterministic consequences of our results, we prove the existence of sparse graphs having the above Ramsey property as well as the existence of infinitely many critical graphs with respect to the property  ...  We consider the binomial random graph G n,p and determine a sharp threshold function for the edge-Ramsey property G n,p → (C 1 , . . . , C r ) for all 1 , . . . , r , where C denotes the cycle of length  ...  When the value of ε is not relevant, we refer to an (ε, k)-equitable partition as a k-equitable partition. Similarly, P is an equitable partition of V if it is a k-equitable partition for some k.  ... 
doi:10.1002/(sici)1098-2418(199710)11:3<245::aid-rsa3>3.3.co;2-i fatcat:oz5dec56zrf33ldcbtdfhuieby

Defective and Clustered Graph Colouring [article]

David R. Wood
2018 arXiv   pre-print
A colouring has "defect" d if each monochromatic component has maximum degree at most d. A colouring has "clustering" c if each monochromatic component has at most c vertices.  ...  Consider the following two ways to colour the vertices of a graph where the requirement that adjacent vertices get distinct colours is relaxed.  ...  I am particularly thankful to Frédéric Havet for allowing his unpublished proof of Lemma 29 to be included.  ... 
arXiv:1803.07694v1 fatcat:prcytiquyzb77cuxaexdflx5lm
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