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Sparse H -Colourable Graphs of Bounded Maximum Degree

Xuding Zhu, Hossein Hajiabolhassan
2004 Graphs and Combinatorics  
We prove that for any integer g there is a 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 .  ... 
doi:10.1007/s00373-003-0542-z fatcat:ppedad6ipbgsxelos7huclfnca

Paths between colourings of sparse graphs [article]

Carl Feghali
2020 arXiv   pre-print
For every ϵ > 0 and every 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.  ... 
arXiv:1803.03950v2 fatcat:3dqnkgvrbzatdm2r3fsf2v45zq

Colourings with Bounded Monochromatic Components in Graphs of Given Circumference [article]

Bojan Mohar and Bruce Reed and David R. Wood
2017 arXiv   pre-print
The O( k) 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  ... 
arXiv:1612.05674v2 fatcat:b2kbmprb2rhxnmycq7wmodeuoy

Colouring locally sparse graphs with the first moment method [article]

François Pirot, Eoin Hurley
2021 arXiv   pre-print
As a final touch, we show that our method provides an asymptotically tight lower 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.  ... 
arXiv:2109.15215v3 fatcat:rxhs3zj3l5c5flapzhltfczn7u

On randomly colouring locally sparse graphs

Alan Frieze, Juan Vera
2006 Discrete Mathematics & Theoretical Computer Science  
We show that if for all v ∈ V the average 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.  ... 
doi:10.46298/dmtcs.360 fatcat:oodwtcfimzcu3hz2fcuvglklc4

Defective and clustered choosability of sparse graphs

Kevin Hendrey, David R. Wood
2019 Combinatorics, probability & computing  
For clustered choosability 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).  ... 
doi:10.1017/s0963548319000063 fatcat:z4w4ci523rfe5ketzigqyzwphi

Colouring planar graphs with bounded monochromatic components

A. N. Glebov
2020 Sibirskie Elektronnye Matematicheskie Izvestiya  
This implies that planar 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.  ... 
doi:10.33048/semi.2020.17.032 fatcat:6ttcqtztnjacfcqgmfzwr6k4uy

Colouring Graphs with Sparse Neighbourhoods: Bounds and Applications [article]

Marthe Bonamy, Thomas Perrett, Luke Postle
2018 arXiv   pre-print
Let G be a 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).  ... 
arXiv:1810.06704v1 fatcat:vy6gbwyjuve4nhuzttlg3kr75e

The degree-diameter problem for sparse graph classes [article]

Guillermo Pineda-Villavicencio, David R. Wood
2014 arXiv   pre-print
For 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 ∆.  ... 
arXiv:1307.4456v3 fatcat:g4wnt4fkvjdzjl67jb4v3pzhb4

Local resilience of spanning subgraphs in sparse random graphs

Peter Allen, Julia Böttcher, Julia Ehrenmüller, Anusch Taraz
2015 Electronic Notes in Discrete Mathematics  
-even after an adversary deletes an arbitrary (1/k − γ)-fraction 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.  ... 
doi:10.1016/j.endm.2015.06.071 fatcat:72iwiorudff2diflg4nsj2yjlq

An improved procedure for colouring graphs of bounded local density [article]

Eoin Hurley and Rémi de Joannis de Verclos and Ross J. Kang
2020 arXiv   pre-print
We develop an improved 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.  ... 
arXiv:2007.07874v2 fatcat:oigdu4znvferfiml7wyvbszzty

An approximate blow-up lemma for sparse pseudorandom graphs

Peter Allen, Julia Böttcher, Hiệp Hàn, Yoshiharu Kohayakawa, Yury Person
2013 Electronic Notes in Discrete Mathematics  
We state a 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.  ... 
doi:10.1016/j.endm.2013.10.061 fatcat:unehq3cnibfkva4s5q5tfit7xy

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.  ... 

Hypergraph Packing and Sparse Bipartite Ramsey Numbers

DAVID CONLON
2009 Combinatorics, probability & computing  
We prove that there exists a constant c such that, for any integer ∆, the Ramsey number 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 .  ... 
doi:10.1017/s0963548309990174 fatcat:eufjgpnn4ve73eiq2dgc7l5rey

The Degree-Diameter Problem for Sparse Graph Classes

Guillermo Pineda-Villavicencio, David R. Wood
2015 Electronic Journal of Combinatorics  
We consider the 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 ∆.  ... 
doi:10.37236/4313 fatcat:ogfi5ymfi5hvhc2rym6qaprd2y
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