11,022 research outputs found
Generalized Colorings of Graphs
A graph coloring is an assignment of labels called “colors” to certain elements of a graph subject to certain constraints. The proper vertex coloring is the most common type of graph coloring, where each vertex of a graph is assigned one color such that no two adjacent vertices share the same color, with the objective of minimizing the number of colors used. One can obtain various generalizations of the proper vertex coloring problem, by strengthening or relaxing the constraints or changing the objective. We study several types of such generalizations in this thesis. Series-parallel graphs are multigraphs that have no K4-minor. We provide bounds on their fractional and circular chromatic numbers and the defective version of these pa-rameters. In particular we show that the fractional chromatic number of any series-parallel graph of odd girth k is exactly 2k/(k − 1), confirming a conjecture by Wang and Yu. We introduce a generalization of defective coloring: each vertex of a graph is assigned a fraction of each color, with the total amount of colors at each vertex summing to 1. We define the fractional defect of a vertex v to be the sum of the overlaps with each neighbor of v, and the fractional defect of the graph to be the maximum of the defects over all vertices. We provide results on the minimum fractional defect of 2-colorings of some graphs. We also propose some open questions and conjectures. Given a (not necessarily proper) vertex coloring of a graph, a subgraph is called rainbow if all its vertices receive different colors, and monochromatic if all its vertices receive the same color. We consider several types of coloring here: a no-rainbow-F coloring of G is a coloring of the vertices of G without rainbow subgraph isomorphic to F ; an F -WORM coloring of G is a coloring of the vertices of G without rainbow or monochromatic subgraph isomorphic to F ; an (M, R)-WORM coloring of G is a coloring of the vertices of G with neither a monochromatic subgraph isomorphic to M nor a rainbow subgraph isomorphic to R. We present some results on these concepts especially with regards to the existence of colorings, complexity, and optimization within certain graph classes. Our focus is on the case that F , M or R is a path, cycle, star, or clique
The Relaxed Edge-Coloring Game and \u3cem\u3ek\u3c/em\u3e-Degenerate Graphs
The (r, d)-relaxed edge-coloring game is a two-player game using r colors played on the edge set of a graph G. We consider this game on forests and more generally, on k-degenerate graphs. If F is a forest with ∆(F) = ∆, then the first player, Alice, has a winning strategy for this game with r = ∆ − j and d ≥ 2j + 2 for 0 ≤ j ≤ ∆ − 1. This both improves and generalizes the result for trees in [10]. More broadly, we generalize the main result in [10] by showing that if G is k-degenerate with ∆(G) = ∆ and j ∈ [∆ + k − 1], then there exists a function h(k, j) such that Alice has a winning strategy for this game with r = ∆ + k − j and d ≥ h(k, j)
The Relaxed Game Chromatic Index of \u3cem\u3ek\u3c/em\u3e-Degenerate Graphs
The (r, d)-relaxed coloring game is a two-player game played on the vertex set of a graph G. We consider a natural analogue to this game on the edge set of G called the (r, d)-relaxed edge-coloring game. We consider this game on trees and more generally, on k-degenerate graphs. We show that if G is k-degenerate with ∆(G) = ∆, then the first player, Alice, has a winning strategy for this game with r = ∆+k−1 and d≥2k2 + 4k
Distributed Deterministic Edge Coloring using Bounded Neighborhood Independence
We study the {edge-coloring} problem in the message-passing model of
distributed computing. This is one of the most fundamental and well-studied
problems in this area. Currently, the best-known deterministic algorithms for
(2Delta -1)-edge-coloring requires O(Delta) + log-star n time \cite{PR01},
where Delta is the maximum degree of the input graph. Also, recent results of
\cite{BE10} for vertex-coloring imply that one can get an
O(Delta)-edge-coloring in O(Delta^{epsilon} \cdot \log n) time, and an
O(Delta^{1 + epsilon})-edge-coloring in O(log Delta log n) time, for an
arbitrarily small constant epsilon > 0.
In this paper we devise a drastically faster deterministic edge-coloring
algorithm. Specifically, our algorithm computes an O(Delta)-edge-coloring in
O(Delta^{epsilon}) + log-star n time, and an O(Delta^{1 +
epsilon})-edge-coloring in O(log Delta) + log-star n time. This result improves
the previous state-of-the-art {exponentially} in a wide range of Delta,
specifically, for 2^{Omega(\log-star n)} \leq Delta \leq polylog(n). In
addition, for small values of Delta our deterministic algorithm outperforms all
the existing {randomized} algorithms for this problem.
On our way to these results we study the {vertex-coloring} problem on the
family of graphs with bounded {neighborhood independence}. This is a large
family, which strictly includes line graphs of r-hypergraphs for any r = O(1),
and graphs of bounded growth. We devise a very fast deterministic algorithm for
vertex-coloring graphs with bounded neighborhood independence. This algorithm
directly gives rise to our edge-coloring algorithms, which apply to {general}
graphs.
Our main technical contribution is a subroutine that computes an
O(Delta/p)-defective p-vertex coloring of graphs with bounded neighborhood
independence in O(p^2) + \log-star n time, for a parameter p, 1 \leq p \leq
Delta
Information-Sharing and Privacy in Social Networks
We present a new model for reasoning about the way information is shared
among friends in a social network, and the resulting ways in which it spreads.
Our model formalizes the intuition that revealing personal information in
social settings involves a trade-off between the benefits of sharing
information with friends, and the risks that additional gossiping will
propagate it to people with whom one is not on friendly terms. We study the
behavior of rational agents in such a situation, and we characterize the
existence and computability of stable information-sharing networks, in which
agents do not have an incentive to change the partners with whom they share
information. We analyze the implications of these stable networks for social
welfare, and the resulting fragmentation of the social network
Defective 3-Paintability of Planar Graphs
A -defective -painting game on a graph is played by two players:
Lister and Painter. Initially, each vertex is uncolored and has tokens. In
each round, Lister marks a chosen set of uncolored vertices and removes one
token from each marked vertex. In response, Painter colors vertices in a subset
of which induce a subgraph of maximum degree at most . Lister
wins the game if at the end of some round there is an uncolored vertex that has
no more tokens left. Otherwise, all vertices eventually get colored and Painter
wins the game. We say that is -defective -paintable if Painter has a
winning strategy in this game. In this paper we show that every planar graph is
3-defective 3-paintable and give a construction of a planar graph that is not
2-defective 3-paintable.Comment: 21 pages, 11 figure
Exponential algorithmic speedup by quantum walk
We construct an oracular (i.e., black box) problem that can be solved
exponentially faster on a quantum computer than on a classical computer. The
quantum algorithm is based on a continuous time quantum walk, and thus employs
a different technique from previous quantum algorithms based on quantum Fourier
transforms. We show how to implement the quantum walk efficiently in our
oracular setting. We then show how this quantum walk can be used to solve our
problem by rapidly traversing a graph. Finally, we prove that no classical
algorithm can solve this problem with high probability in subexponential time.Comment: 24 pages, 7 figures; minor corrections and clarification
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
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