272 research outputs found
Brooks-type theorem for -hued coloring of graphs
An -hued coloring of a simple graph is a proper coloring of its
vertices such that every vertex is adjacent to at least differently colored vertices. The minimum number of colors needed
for an -hued coloring of a graph , the -hued chromatic number, is
denoted by . In this note we show that for every simple graph and every , which
in the case when improves the presently known -based
upper bound on , namely .
We also discuss the existence of graphs whose -hued chromatic number is
close to and we prove that there is a bipartite graph
of maximum degree whose -hued chromatic number is
for every and infinitely many values of ; we believe that is the best upper bound on the
-hued chromatic number of any bipartite graph
Deterministic Graph Exploration with Advice
We consider the task of graph exploration. An -node graph has unlabeled
nodes, and all ports at any node of degree are arbitrarily numbered
. A mobile agent has to visit all nodes and stop. The exploration
time is the number of edge traversals. We consider the problem of how much
knowledge the agent has to have a priori, in order to explore the graph in a
given time, using a deterministic algorithm. This a priori information (advice)
is provided to the agent by an oracle, in the form of a binary string, whose
length is called the size of advice. We consider two types of oracles. The
instance oracle knows the entire instance of the exploration problem, i.e., the
port-numbered map of the graph and the starting node of the agent in this map.
The map oracle knows the port-numbered map of the graph but does not know the
starting node of the agent.
We first consider exploration in polynomial time, and determine the exact
minimum size of advice to achieve it. This size is ,
for both types of oracles.
When advice is large, there are two natural time thresholds:
for a map oracle, and for an instance oracle, that can be achieved
with sufficiently large advice. We show that, with a map oracle, time
cannot be improved in general, regardless of the size of advice.
We also show that the smallest size of advice to achieve this time is larger
than , for any .
For an instance oracle, advice of size is enough to achieve time
. We show that, with any advice of size , the time of
exploration must be at least , for any , and with any
advice of size , the time must be .
We also investigate minimum advice sufficient for fast exploration of
hamiltonian graphs
Fully Scalable Massively Parallel Algorithms for Embedded Planar Graphs
We consider the massively parallel computation (MPC) model, which is a
theoretical abstraction of large-scale parallel processing models such as
MapReduce. In this model, assuming the widely believed 1-vs-2-cycles
conjecture, solving many basic graph problems in rounds with a strongly
sublinear memory size per machine is impossible. We improve on the recent work
of Holm and T\v{e}tek [SODA 2023] that bypass this barrier for problems when a
planar embedding of the graph is given. In the previous work, on graphs of size
with machines, the memory size per machine needs to be
at least , whereas we extend their work to the
fully scalable regime, where the memory size per machine can be for any constant . We give the first constant round
fully scalable algorithms for embedded planar graphs for the problems of (i)
connectivity and (ii) minimum spanning tree (MST). Moreover, we show that the
-emulator of Chang, Krauthgamer, and Tan [STOC 2022] can be
incorporated into our recursive framework to obtain constant-round
-approximation algorithms for the problems of computing (iii)
single source shortest path (SSSP), (iv) global min-cut, and (v) -max flow.
All previous results on cuts and flows required linear memory in the MPC model.
Furthermore, our results give new algorithms for problems that implicitly
involve embedded planar graphs. We give as corollaries constant round fully
scalable algorithms for (vi) 2D Euclidean MST using total memory and
(vii) -approximate weighted edit distance using
memory.
Our main technique is a recursive framework combined with novel graph drawing
algorithms to compute smaller embedded planar graphs in constant rounds in the
fully scalable setting.Comment: To appear in SODA24. 55 pages, 9 figures, 1 table. Added section on
weighted edit distance and shortened abstrac
Density and Chromatic Index, and Minimum Ranks of Sign Pattern Matrices
Given a (multi)graph, the density is defined by The {\bf chromatic index} of a graph is the minimum number of colors that required to color the edges of such that two adjacent edges receive different colors. It is known that . The {\bf cover index} of is the greatest integer for which there is a coloring of with colors such that each vertex of is incident with at least one edge of each color. A sign pattern is a matrix whose entries are from the set .
In part 1, we will generally discuss the connections between the density and the chromatic index. In particular, the Goldberg-Seymour conjecture states that if , where is the maximum degree of . Some open problems are mentioned at the end of part 1. In particular, a dual conjecture to the Goldberg-Seymour conjecture on the cover index is discussed. A proof of the Goldberg-Seymour conjecture is given In part 2.
In part 3, we will present a connection between the minimum ranks of sign pattern matrices and point-line configurations
Cops and Robber Game with a Fast Robber
Graph searching problems are described as games played on graphs, between a set of searchers and a fugitive. Variants of the game restrict the abilities of the searchers and the fugitive and the corresponding search number (the least number of searchers that have a winning strategy) is related to several well-known parameters in graph theory. One popular variant is called the Cops and Robber game, where the searchers (cops) and the fugitive (robber) move in rounds, and in each round they move to an adjacent vertex. This game, defined in late 1970's, has been studied intensively. The most famous open problem is Meyniel's conjecture, which states that the cop number
(the minimum number of cops that can always capture the robber) of a connected graph
on n vertices is O(sqrt n).
We consider a version of the Cops and Robber game, where the robber is faster than the cops, but is not allowed to jump over the cops. This version was first studied in 2008.
We show that when the robber has speed s,
the cop number of a connected n-vertex graph can be as large as Omega(n^(s/s+1)). This improves the Omega(n^(s-3/s-2)) lower bound of Frieze, Krivelevich, and Loh (Variations on Cops and Robbers, J. Graph Theory, to appear). We also conjecture a general upper bound O(n^(s/s+1)) for the cop number,
generalizing Meyniel's conjecture.
Then we focus on the version where the robber is infinitely fast, but is again not allowed to jump over the cops. We give a mathematical characterization for graphs with cop number one. For a graph with treewidth tw and maximum degree Delta,
we prove the cop number is between (tw+1)/(Delta+1) and tw+1. Using this we show that the cop number of the m-dimensional hypercube is
between c1 n / m sqrt(m) and c2 n / m for some constants c1 and c2. If G is a connected interval graph on n vertices, then we give a polynomial time 3-approximation algorithm for finding the cop number of G, and prove that the cop number is O(sqrt(n)).
We prove that given n, there exists a connected chordal graph on n vertices
with cop number Omega(n/log n). We show a lower bound for the cop numbers of expander graphs, and use this to prove that the random G(n,p) that is not very sparse,
asymptotically almost surely has cop number between d1 / p and d2 log (np) / p for suitable constants d1 and d2. Moreover, we prove that a fixed-degree regular random graph with n vertices asymptotically almost surely has cop number Theta(n)
Property testing of graphs and the role of neighborhood distributions
Property testing considers decision problems in the regime of sublinear complexity. Most classical decision problems require at least linear time complexity in order to read the whole input. Hence, decision problems are relaxed by introducing a gap between “yes” and “no” instances: A property tester for a property Π (e. g., planarity) is a randomized algorithm with constant error probability that accepts objects that have Π (planar graphs) and that rejects objects that have linear edit distance to any object from Π (graphs with a linear number of crossing edges in every planar embedding). For property testers, locality is a natural and crucial concept because they cannot obtain a global view of their input. In this thesis, we investigate property testing in graphs and how testers leverage the information contained in the neighborhoods of randomly sampled vertices: We provide some structural insights regarding properties with constant testing complexity in graphs with bounded (maximum vertex) degree and a connection between testers with constant complexity for general graphs and testers with logarithmic space complexity for random-order streams. We also present testers for some minor-freeness properties and a tester for conductance in the distributed CONGEST model
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