7,453 research outputs found
Relaxed Coloring of Sparse Graphs
A graph G is (d_1,d_2,… ,d_t)-colorable if its vertices may be partitioned into subsets V_1,V_2,...,V_t such that for each i, the maximum degree of the subgraph induced by V_i is at most d_i. We study this relaxed coloring of graphs with bounded maximum average degrees. Specifically, we use discharging and other methods to seek new upper and lower bounds for the maximum average degree of (1,1,0)-colorable graphs. We generalize this result to colorings of the type (1_1,1_2,...,1_a,0_1,...,0_b), improving the results by Dorbec, Kaiser, Montassier, and Raspaud (J. of Graph Theory, 2014) for a large class of colorings
Graph coloring with no large monochromatic components
For a graph G and an integer t we let mcc_t(G) be the smallest m such that
there exists a coloring of the vertices of G by t colors with no monochromatic
connected subgraph having more than m vertices. Let F be any nontrivial
minor-closed family of graphs. We show that \mcc_2(G) = O(n^{2/3}) for any
n-vertex graph G \in F. This bound is asymptotically optimal and it is attained
for planar graphs. More generally, for every such F and every fixed t we show
that mcc_t(G)=O(n^{2/(t+1)}). On the other hand we have examples of graphs G
with no K_{t+3} minor and with mcc_t(G)=\Omega(n^{2/(2t-1)}).
It is also interesting to consider graphs of bounded degrees. Haxell, Szabo,
and Tardos proved \mcc_2(G) \leq 20000 for every graph G of maximum degree 5.
We show that there are n-vertex 7-regular graphs G with \mcc_2(G)=\Omega(n),
and more sharply, for every \epsilon>0 there exists c_\epsilon>0 and n-vertex
graphs of maximum degree 7, average degree at most 6+\epsilon for all
subgraphs, and with mcc_2(G)\ge c_\eps n. For 6-regular graphs it is known only
that the maximum order of magnitude of \mcc_2 is between \sqrt n and n.
We also offer a Ramsey-theoretic perspective of the quantity \mcc_t(G).Comment: 13 pages, 2 figure
Defective and Clustered Graph Colouring
Consider the following two ways to colour the vertices of a graph where the
requirement that adjacent vertices get distinct colours is relaxed. A colouring
has "defect" if each monochromatic component has maximum degree at most
. A colouring has "clustering" if each monochromatic component has at
most vertices. This paper surveys research on these types of colourings,
where the first priority is to minimise the number of colours, with small
defect or small clustering as a secondary goal. List colouring variants are
also considered. The following graph classes are studied: outerplanar graphs,
planar graphs, graphs embeddable in surfaces, graphs with given maximum degree,
graphs with given maximum average degree, graphs excluding a given subgraph,
graphs with linear crossing number, linklessly or knotlessly embeddable graphs,
graphs with given Colin de Verdi\`ere parameter, graphs with given
circumference, graphs excluding a fixed graph as an immersion, graphs with
given thickness, graphs with given stack- or queue-number, graphs excluding
as a minor, graphs excluding as a minor, and graphs excluding
an arbitrary graph as a minor. Several open problems are discussed.Comment: This is a preliminary version of a dynamic survey to be published in
the Electronic Journal of Combinatoric
Streaming and Massively Parallel Algorithms for Edge Coloring
A valid edge-coloring of a graph is an assignment of "colors" to its edges such that no two incident edges receive the same color. The goal is to find a proper coloring that uses few colors. (Note that the maximum degree, Delta, is a trivial lower bound.) In this paper, we revisit this fundamental problem in two models of computation specific to massive graphs, the Massively Parallel Computations (MPC) model and the Graph Streaming model:
- Massively Parallel Computation: We give a randomized MPC algorithm that with high probability returns a Delta+O~(Delta^(3/4)) edge coloring in O(1) rounds using O(n) space per machine and O(m) total space. The space per machine can also be further improved to n^(1-Omega(1)) if Delta = n^Omega(1). Our algorithm improves upon a previous result of Harvey et al. [SPAA 2018].
- Graph Streaming: Since the output of edge-coloring is as large as its input, we consider a standard variant of the streaming model where the output is also reported in a streaming fashion. The main challenge is that the algorithm cannot "remember" all the reported edge colors, yet has to output a proper edge coloring using few colors.
We give a one-pass O~(n)-space streaming algorithm that always returns a valid coloring and uses 5.44 Delta colors with high probability if the edges arrive in a random order. For adversarial order streams, we give another one-pass O~(n)-space algorithm that requires O(Delta^2) colors
Defective and Clustered Choosability of Sparse Graphs
An (improper) graph colouring has "defect" if each monochromatic subgraph
has maximum degree at most , and has "clustering" if each monochromatic
component has at most vertices. This paper studies defective and clustered
list-colourings for graphs with given maximum average degree. We prove that
every graph with maximum average degree less than is
-choosable with defect . This improves upon a similar result by Havet and
Sereni [J. Graph Theory, 2006]. For clustered choosability of graphs with
maximum average degree , no bound on the number of colours
was previously known. The above result with solves this problem. It
implies that every graph with maximum average degree is
-choosable with clustering 2. This extends a
result of Kopreski and Yu [Discrete Math., 2017] to the setting of
choosability. We then prove two results about clustered choosability that
explore the trade-off between the number of colours and the clustering. In
particular, we prove that every graph with maximum average degree is
-choosable with clustering , and is
-choosable with clustering . As an
example, the later result implies that every biplanar graph is 8-choosable with
bounded clustering. This is the best known result for the clustered version of
the earth-moon problem. The results extend to the setting where we only
consider the maximum average degree of subgraphs with at least some number of
vertices. Several applications are presented
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