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Equitable Colourings of d-degenerate Graphs

2003
*
Combinatorics, probability & computing
*

A proper vertex coloring

doi:10.1017/s0963548302005485
fatcat:fwiz4krpwzcthpupn573hylxwi
*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. ...##
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Equitable d-degenerate Choosability of Graphs
[chapter]

2020
*
Lecture Notes in Computer Science
*

Let

doi:10.1007/978-3-030-48966-3_19
fatcat:uo5s5a24zrcmpegxcdyltuyukm
*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*. ...##
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Equitable d-degenerate choosability of graphs
[article]

2020
*
arXiv
*
pre-print

Let D_d be the class

arXiv:2003.09722v1
fatcat:4y4jbnhbmra3hepky7b44jod7m
*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*. ...##
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Equitable list vertex colourability and arboricity of grids

2018
*
Filomat
*

A

doi:10.2298/fil1818353d
fatcat:2wc26mgl3rgn3moxkdnphwoqc4
*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*...##
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Equitable List Vertex Colourability and Arboricity of Grids
[article]

2018
*
arXiv
*
pre-print

A

arXiv:1809.08281v1
fatcat:auwso2nbofh6hnoaz7du53nvfq
*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. ...##
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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. ...##
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Independent Coverings and Orthogonal Colourings
[article]

2021
*
arXiv
*
pre-print

One conjecture regards independent coverings

arXiv:2008.07904v3
fatcat:seagrwz7fvgqrbc3ba4af3o3va
*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*. ...##
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Equitable colorings of $K_4$-minor-free graphs

2017
*
Journal of Graph Algorithms and Applications
*

We do not use the discharging method but rather exploit decomposition trees

doi:10.7155/jgaa.00451
fatcat:kp2bugfz7ba77mmgu2ym7numgm
*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. ...##
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Equitable Colorings of K_4-minor-free Graphs
[article]

2017
*
arXiv
*
pre-print

We do not use the discharging method but rather exploit decomposition trees

arXiv:1703.02250v1
fatcat:pjlsrohssnf2zas4x5cduxthoq
*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. ...##
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Spanning embeddings of arrangeable graphs with sublinear bandwidth
[article]

2013
*
arXiv
*
pre-print

We generalise this result to a-arrangeable

arXiv:1305.2078v1
fatcat:fxskstnyffcw3cymldtgddmij4
*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. ...##
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Spanning embeddings of arrangeable graphs with sublinear bandwidth

2015
*
Random structures & algorithms (Print)
*

We generalise this result to a-arrangeable

doi:10.1002/rsa.20593
fatcat:doc423tz7neqfjbiorduj44uey
*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. ...##
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The Bandwidth Theorem in Sparse Graphs
[article]

2020
*
arXiv
*
pre-print

More precisely, we show that for p≫(log nn)^1/Δ asymptotically almost surely each spanning subgraph G

arXiv:1612.00661v2
fatcat:5etgg2uiqrapped3aqdegi4ufy
*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 ...##
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Towards Lehel's conjecture for 4-uniform tight cycles
[article]

2020
*
arXiv
*
pre-print

A k-uniform tight cycle is a k-uniform hypergraph with a cyclic ordering

arXiv:2012.08875v1
fatcat:qgrk526snbhonfrburywvollpy
*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 ...##
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Threshold functions for asymmetric Ramsey properties involving cycles

1997
*
Random structures & algorithms (Print)
*

As deterministic consequences

doi:10.1002/(sici)1098-2418(199710)11:3<245::aid-rsa3>3.3.co;2-i
fatcat:oz5dec56zrf33ldcbtdfhuieby
*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. ...##
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Defective and Clustered Graph Colouring
[article]

2018
*
arXiv
*
pre-print

A

arXiv:1803.07694v1
fatcat:prcytiquyzb77cuxaexdflx5lm
*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. ...
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