43 research outputs found
Note on 3-Coloring of -Free Graphs
We show that the 3-coloring problem is polynomial-time solvable on
-free graphs.Comment: 17 pages, 13 figure
Color-blind index in graphs of very low degree
Let be an edge-coloring of a graph , not necessarily
proper. For each vertex , let , where is
the number of edges incident to with color . Reorder for
every in in nonincreasing order to obtain , the color-blind
partition of . When induces a proper vertex coloring, that is,
for every edge in , we say that is color-blind
distinguishing. The minimum for which there exists a color-blind
distinguishing edge coloring is the color-blind index of ,
denoted . We demonstrate that determining the
color-blind index is more subtle than previously thought. In particular,
determining if is NP-complete. We also connect
the color-blind index of a regular bipartite graph to 2-colorable regular
hypergraphs and characterize when is finite for a class
of 3-regular graphs.Comment: 10 pages, 3 figures, and a 4 page appendi
List precoloring extension in planar graphs
A celebrated result of Thomassen states that not only can every planar graph
be colored properly with five colors, but no matter how arbitrary palettes of
five colors are assigned to vertices, one can choose a color from the
corresponding palette for each vertex so that the resulting coloring is proper.
This result is referred to as 5-choosability of planar graphs. Albertson asked
whether Thomassen's theorem can be extended by precoloring some vertices which
are at a large enough distance apart in a graph. Here, among others, we answer
the question in the case when the graph does not contain short cycles
separating precolored vertices and when there is a "wide" Steiner tree
containing all the precolored vertices.Comment: v2: 15 pages, 11 figres, corrected typos and new proof of Theorem
3(2
Graphs with tiny vector chromatic numbers and huge chromatic numbers
Karger, Motwani, and Sudan [J. ACM, 45 (1998), pp. 246-265] introduced the notion of a vector coloring of a graph. In particular, they showed that every k-colorable graph is also vector k-colorable, and that for constant k, graphs that are vector k-colorable can be colored by roughly Δ^(1 - 2/k) colors. Here Δ is the maximum degree in the graph and is assumed to be of the order of n^5 for some 0 < δ < 1. Their results play a major role in the best approximation algorithms used for coloring and for maximum independent sets. We show that for every positive integer k there are graphs that are vector k-colorable but do not have independent sets significantly larger than n/Δ^(1- 2/k) (and hence cannot be colored with significantly fewer than Δ^(1-2/k) colors). For k = O(log n/log log n) we show vector k-colorable graphs that do not have independent sets of size (log n)^c, for some constant c. This shows that the vector chromatic number does not approximate the chromatic number within factors better than n/polylogn.
As part of our proof, we analyze "property testing" algorithms that distinguish between graphs that have an independent set of size n/k, and graphs that are "far" from having such an independent set. Our bounds on the sample size improve previous bounds of Goldreich, Goldwasser, and Ron [J. ACM, 45 (1998), pp. 653-750] for this problem
Об индексе палитры треугольника Серпинского и графа Серпинского
The palette of a vertex v of a graph G in a proper edge coloring is the set of colors assigned to the edges which are incident to v. The palette index of G is the minimum number of palettes occurring among all proper edge colorings of G. In this paper, we consider the palette index of Sierpinski graphs S” and Sierpinski triangle graphs S” . In particular, we determine the exact value of the palette index of Sierpinski triangle graphs. We also determine the palette index of Sierpinski graphs S” where p is even, p = 3, or n = 2 and p = 41 + 3
No distributed quantum advantage for approximate graph coloring
We give an almost complete characterization of the hardness of -coloring
-chromatic graphs with distributed algorithms, for a wide range of models
of distributed computing. In particular, we show that these problems do not
admit any distributed quantum advantage. To do that: 1) We give a new
distributed algorithm that finds a -coloring in -chromatic graphs in
rounds, with . 2) We prove that any distributed
algorithm for this problem requires rounds.
Our upper bound holds in the classical, deterministic LOCAL model, while the
near-matching lower bound holds in the non-signaling model. This model,
introduced by Arfaoui and Fraigniaud in 2014, captures all models of
distributed graph algorithms that obey physical causality; this includes not
only classical deterministic LOCAL and randomized LOCAL but also quantum-LOCAL,
even with a pre-shared quantum state.
We also show that similar arguments can be used to prove that, e.g.,
3-coloring 2-dimensional grids or -coloring trees remain hard problems even
for the non-signaling model, and in particular do not admit any quantum
advantage. Our lower-bound arguments are purely graph-theoretic at heart; no
background on quantum information theory is needed to establish the proofs
List-coloring and sum-list-coloring problems on graphs
Graph coloring is a well-known and well-studied area of graph theory that has many applications. In this dissertation, we look at two generalizations of graph coloring known as list-coloring and sum-list-coloring. In both of these types of colorings, one seeks to first assign palettes of colors to vertices and then choose a color from the corresponding palette for each vertex so that a proper coloring is obtained.
A celebrated result of Thomassen states that every planar graph can be properly colored from any arbitrarily assigned palettes of five colors. This result is known as 5-list-colorability of planar graphs. Albertson asked whether Thomassen\u27s theorem can be extended by precoloring some vertices which are at a large enough distance apart. Hutchinson asked whether Thomassen\u27s theorem can be extended by allowing certain vertices to have palettes of size less than five assigned to them. In this dissertation, we explore both of these questions and answer them in the affirmative for various classes of graphs.
We also provide a catalog of small configurations with palettes of different prescribed sizes and determine whether or not they can always be colored from palettes of such sizes. These small configurations can be useful in reducing certain planar graphs to obtain more information about their structure.
Additionally, we look at the newer notion of sum-list-coloring where the sum choice number is the parameter of interest. In sum-list-coloring, we seek to minimize the sum of varying sizes of palettes of colors assigned the vertices of a graph. We compute the sum choice number for all graphs on at most five vertices, present some general results about sum-list-coloring, and determine the sum choice number for certain graphs made up of cycles