• Corpus ID: 237605411

A special case of Vu's conjecture: Coloring nearly disjoint graphs of bounded maximum degree

  title={A special case of Vu's conjecture: Coloring nearly disjoint graphs of bounded maximum degree},
  author={Tom Kelly and Daniela K{\"u}hn and Deryk Osthus},
A collection of graphs is nearly disjoint if every pair of them intersects in at most one vertex. We prove that if G1, . . . , Gm are nearly disjoint graphs of maximum degree at most D, then the following holds. For every fixed C, if each vertex v ∈ ⋃m i=1 V (Gi) is contained in at most C of the graphs G1, . . . , Gm, then the (list) chromatic number of ⋃m i=1 Gi is at most D + o(D). This result confirms a special case of a conjecture of Vu and generalizes Kahn’s bound on the list chromatic… 
Solution to a problem of Erd\H{o}s on the chromatic index of hypergraphs with bounded codegree
In 1977, Erdős asked the following question: for any integers t, n ∈ N, if G1, . . . , Gn are complete graphs such that each Gi has at most n vertices and every pair of them shares at most t
Graph and hypergraph colouring via nibble methods: A survey
A survey of methods, results, and open problems on graph and hypergraph colourings, with a particular emphasis on semi-random ‘nibble’ methods.


Asymptotic behavior of the chromatic index for hypergraphs
The chromatic index of a Steiner triple-system on n points is asymptotic to n 2, resolving a long-standing conjecture and strengthening and generalizing a result due to Frankl and Rodl concerning the existence of a single almost perfect packing or covering under similar circumstances.
A General Upper Bound on the List Chromatic Number of Locally Sparse Graphs
  • V. Vu
  • Computer Science, Mathematics
    Combinatorics, Probability and Computing
  • 2002
Several upper bounds for the strong (list) chromatic index of a graph are derived, under various assumptions, and this result is sharp up to the multiplicative constant K and strengthens previous results by Kim, Johansson, Alon, Krivelevich and Sudakov.
Asymptotically Good List-Colorings
  • J. Kahn
  • Mathematics, Computer Science
    J. Comb. Theory, Ser. A
  • 1996
The “guided-random” method used in the proof is in the spirit of some earlier work and is thought to be of particular interest, and one simple ingredient is a martingale inequality which ought to prove useful beyond the present context.
Coloring graphs with forbidden bipartite subgraphs
A conjecture that, for any graph F, there is a constant cF > 0 such that if G is an F -free graph of maximum degree ∆, then χ(G) 6 cF∆/ log ∆ is verified for a class of graphs F that includes all bipartite graphs.
Problems and Results on 3-chromatic Hypergraphs and Some Related Questions
A hypergraphi is a collection of sets. This paper deals with finite hy-pergraphs only. The sets in the hypergraph are called edges, the elements of these edges are points. The degree of a point is
Asymptotically the List Colouring Constants Are 1
In this paper we prove the following result about vertex list colourings, which shows that a conjecture of the first author (1999, J. Graph Theory 31, 149-153) is asymptotically correct. Let G be a
An average degree condition for independent transversals
In 1994, Erdős, Gyarfas and Łuczak posed the following problem: given disjoint vertex sets $V_1,\dots,V_n$ of size~$k$, with exactly one edge between any pair $V_i,V_j$, how large can $n$ be such
A Note on Vertex List Colouring
  • P. Haxell
  • Mathematics, Computer Science
    Combinatorics, Probability and Computing
  • 2001
It is proved that there exists a proper vertex colouring f of G such that f( v) ∈ S(v) for each v ∈ V(G) and this proves a weak version of a conjecture of Reed.
On a list coloring conjecture of Reed
We construct graphs with lists of available colors for each vertex, such that the size of every list exceeds the maximum vertex-color degree, but there exists no proper coloring from the lists. This
Palette Sparsification Beyond (Δ+1) Vertex Coloring
This paper proves that sampling Oε(logn) colors per vertex is sufficient for proper coloring of any graph with high probability whenever each vertex is sampling from a list of (1 + ε) · deg(v) arbitrary colors, or even only deg( v) + 1 colors when the lists are the sets of sets of colors.