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Sparse H -Colourable Graphs of Bounded Maximum Degree

2004
*
Graphs and Combinatorics
*

We prove that for any integer g there is a

doi:10.1007/s00373-003-0542-z
fatcat:ppedad6ipbgsxelos7huclfnca
*graph*G*of*girth at least g and*of**maximum**degree*at most 5k 13 such that G admits a surjective homomorphism c to F , and moreover, for any F -pointed*graph**H*... and*of**maximum**degree*at most 5k 26mt (where m = |X|) such that X ⊆ V (G) and up to an automorphism*of**H*, there are exactly t homomorphisms from G to*H*, each*of*which is an extension*of*an f ∈ F. ... Recently, it is proved in [1] that there exist uniquely k-*colourable**graphs*G*of*large girth with*maximum**degree*∆(G) ≤ 5k 13 . ...##
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Paths between colourings of sparse graphs
[article]

2020
*
arXiv
*
pre-print

For every ϵ > 0 and every

arXiv:1803.03950v2
fatcat:3dqnkgvrbzatdm2r3fsf2v45zq
*graph*G with n vertices and*maximum*average*degree*d - ϵ, there exists a constant c = c(d, ϵ) such that R_k(G) has diameter O(n^c). ... The reconfiguration*graph*R_k(G)*of*the k-*colourings**of*a*graph*G has as vertex set the set*of*all possible k-*colourings**of*G and two*colourings*are adjacent if they differ on exactly one vertex. ... This work was supported by the research Council*of*Norway via the project CLASSIS, grant number 249994. ...##
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Colourings with Bounded Monochromatic Components in Graphs of Given Circumference
[article]

2017
*
arXiv
*
pre-print

The O( k)

arXiv:1612.05674v2
fatcat:b2kbmprb2rhxnmycq7wmodeuoy
*bound*on the number*of**colours*is best possible, even in the setting*of**colourings*with*bounded*monochromatic*degree*. ... We prove that every*graph*with circumference at most k is O( k)-*colourable*such that every monochromatic component has size at most O(k). ... For a*graph**H*, let f (*H*) be the minimum integer c such that there exists an integer d such that every*H*-minor-free*graph*has a c-*colouring*in which every monochromatic component has*maximum**degree*at most ...##
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Colouring locally sparse graphs with the first moment method
[article]

2021
*
arXiv
*
pre-print

As a final touch, we show that our method provides an asymptotically tight lower

arXiv:2109.15215v3
fatcat:rxhs3zj3l5c5flapzhltfczn7u
*bound*on the number*of**colourings**of*locally*sparse**graphs*. ... We give a short proof*of*a*bound*on the list chromatic number*of**graphs*G*of**maximum**degree*Δ where each neighbourhood has density at most d, namely χ_ℓ(G) ≤ (1+o(1)) Δ/lnΔ/d+1 as Δ/d+1→∞. ... Our main result considers list*colourings**of**graphs**of**maximum**degree*∆ where each neighbourhood induces a subgraph*of**maximum**degree*d. Theorem 1. ...##
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On randomly colouring locally sparse graphs

2006
*
Discrete Mathematics & Theoretical Computer Science
*

We show that if for all v ∈ V the average

doi:10.46298/dmtcs.360
fatcat:oodwtcfimzcu3hz2fcuvglklc4
*degree**of*the subgraph H_v induced by the neighbours*of*v ∈ V is #x226a Δ where Δ is the*maximum**degree*and Δ >c_1\ln n then for sufficiently large c_1, this chain ... For this class*of**graphs*, which includes planar*graphs*, triangle free*graphs*and random*graphs*G_\n,p\ with p #x226a 1, this beats the 11Δ /6*bound**of*Vigoda for general*graphs*. ... Dyer, Flaxman, Frieze and Vigoda [4] show that for*sparse*random*graphs*, the number*of**colours*required for rapid mixing is*of*order the average rather than*maximum**degree*whp. ...##
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Defective and clustered choosability of sparse graphs

2019
*
Combinatorics, probability & computing
*

For clustered choosability

doi:10.1017/s0963548319000063
fatcat:z4w4ci523rfe5ketzigqyzwphi
*of**graphs*with*maximum*average*degree*m, no (1-ɛ)m*bound*on the number*of**colours*was previously known. The above result with d=1 solves this problem. ... This paper studies defective and clustered list-*colourings*for*graphs*with given*maximum*average*degree*. ... This research was initiated at the Bellairs Workshop on*Graph*Theory (20-27 April 2018). ...##
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Colouring planar graphs with bounded monochromatic components

2020
*
Sibirskie Elektronnye Matematicheskie Izvestiya
*

This implies that planar

doi:10.33048/semi.2020.17.032
fatcat:6ttcqtztnjacfcqgmfzwr6k4uy
*graphs**of*girth 5, 6, and 8 are 2-choosable so that each monochromatic component is a tree with*maximum**degree*at most 4, 2, and 1, respectively. ... Finally, we prove that every*graph*with fractional arboricity at most 2d+2 d+2 is 2-choosabale with the property that each monochromatic component is a tree with*maximum**degree*at most d. ... So, the natural mesaure*of**sparseness**of*a*graph*here would be its fractional arboricity rather than the*maximum*average*degree*. Let*H*= (V*H*, E*H*) be an arbitrary subgraph*of*a*graph*G. ...##
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Colouring Graphs with Sparse Neighbourhoods: Bounds and Applications
[article]

2018
*
arXiv
*
pre-print

Let G be a

arXiv:1810.06704v1
fatcat:vy6gbwyjuve4nhuzttlg3kr75e
*graph*with chromatic number χ,*maximum**degree*Δ and clique number ω. Reed's conjecture states that χ≤ (1-ε)(Δ + 1) + εω for all ε≤ 1/2. ... We derive this result from a general technique to*bound*the chromatic number*of*a*graph*where no vertex has many edges in its neighbourhood. ... Lemma 4 . 2 . 42 Let*H*be a*graph**of**maximum**degree*∆, and G = L 2 (*H*). ...##
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The degree-diameter problem for sparse graph classes
[article]

2014
*
arXiv
*
pre-print

For

arXiv:1307.4456v3
fatcat:g4wnt4fkvjdzjl67jb4v3pzhb4
*graphs**of**bounded*average*degree*the answer is Θ(Δ^k-1), and for*graphs**of**bounded*arboricity the answer is Θ(Δ^k/2), in both cases for fixed k. ... The*degree*-diameter problem asks for the*maximum*number*of*vertices in a*graph*with*maximum**degree*Δ and diameter k. For fixed k, the answer is Θ(Δ^k). ... Canale and Gómez [3] established the best known asymptotic*bound**of*N (∆, k) some particular classes G*of**sparse**graphs*, focusing on the case*of*small diameter k, and large*maximum**degree*∆. ...##
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Local resilience of spanning subgraphs in sparse random graphs

2015
*
Electronic Notes in Discrete Mathematics
*

-even after an adversary deletes an arbitrary (1/k − γ)-fraction

doi:10.1016/j.endm.2015.06.071
fatcat:72iwiorudff2diflg4nsj2yjlq
*of*the edges at every vertex -a copy*of*every nvertex*graph*with*maximum**degree*at most Δ, bandwidth at most βn and at least C max{p −2 ... For each real γ > 0 and integers Δ ≥ 2 and k ≥ 1, we prove that there exist constants β > 0 and C > 0 such that for all p ≥ C(log n/n) 1/Δ the random*graph*G(n, p) asymptotically almost surely contains ... The lemma is too long and complicated to be stated here in detail, but it serves as a powerful tool for embedding*maximum**degree**bounded*spanning*graphs*into*sparse**graphs*. ...##
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An improved procedure for colouring graphs of bounded local density
[article]

2020
*
arXiv
*
pre-print

We develop an improved

arXiv:2007.07874v2
fatcat:oigdu4znvferfiml7wyvbszzty
*bound*for the chromatic number*of**graphs**of**maximum**degree*Δ under the assumption that the number*of*edges spanning any neighbourhood is at most (1-σ)Δ2 for some fixed 0<σ<1. ... The leading term in the reduction*of**colours*achieved through this*bound*is best possible as σ→0. ... Acknowledgements We are grateful to Luke Postle for bringing to our attention two important corrections upon an earlier version*of*this manuscript. ...##
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An approximate blow-up lemma for sparse pseudorandom graphs

2013
*
Electronic Notes in Discrete Mathematics
*

We state a

doi:10.1016/j.endm.2013.10.061
fatcat:unehq3cnibfkva4s5q5tfit7xy
*sparse*approximate version*of*the blow-up lemma, showing that regular partitions in su ciently pseudorandom*graphs*behave almost like complete partite*graphs*for embedding*graphs*with*maximum*...*degree*. ... Suppose*H*is an n-vertex, k-*colourable**graph*with*maximum**degree*at most and bandwidth at most n. ...##
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Page 2088 of Mathematical Reviews Vol. , Issue 98D
[page]

1998
*
Mathematical Reviews
*

A longstanding conjecture

*of*Behzad and Vizing claims that A(G)+1< x"(G) < A(G) +2, where A(G) is the*maximum**degree**of*a vertex in G. The lower*bound*is sharp, the upper*bound*remains to be proved. ... A homogeneous set*of*a*graph*G is a set*H**of*vertices*of*G such that |*H*| > 2,|*H*| <|V(G)|, and every vertex outside*H*is adjacent either to all, or to no, vertices*of**H*. ...##
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Hypergraph Packing and Sparse Bipartite Ramsey Numbers

2009
*
Combinatorics, probability & computing
*

We prove that there exists a constant c such that, for any integer ∆, the Ramsey number

doi:10.1017/s0963548309990174
fatcat:eufjgpnn4ve73eiq2dgc7l5rey
*of*a bipartite*graph*on n vertices with*maximum**degree*∆ is less than 2 c∆ n. ... Our proof hinges upon a quantitative form*of*a hypergraph packing result*of*Rödl, Ruciński and Taraz. ... any*graph**H*= (U, V ; F ) with*maximum**degree*∆ for which |U | ≤ |A | and |V | ≤ β ∆ N . ...##
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The Degree-Diameter Problem for Sparse Graph Classes

2015
*
Electronic Journal of Combinatorics
*

We consider the

doi:10.37236/4313
fatcat:ogfi5ymfi5hvhc2rym6qaprd2y
*degree*-diameter problem for particular classes*of**sparse**graphs*, and establish the following results. ... The*degree*-diameter problem asks for the*maximum*number*of*vertices in a*graph*with*maximum**degree*$\Delta$ and diameter $k$. For fixed $k$, the answer is $\Theta(\Delta^k)$. ... Canale and Gómez [3] established the best known asymptotic*bound**of*N (∆, k) some particular classes G*of**sparse**graphs*, focusing on the case*of*small diameter k, and large*maximum**degree*∆. ...
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