45,949 research outputs found
An algebraic formulation of the graph reconstruction conjecture
The graph reconstruction conjecture asserts that every finite simple graph on
at least three vertices can be reconstructed up to isomorphism from its deck -
the collection of its vertex-deleted subgraphs. Kocay's Lemma is an important
tool in graph reconstruction. Roughly speaking, given the deck of a graph
and any finite sequence of graphs, it gives a linear constraint that every
reconstruction of must satisfy.
Let be the number of distinct (mutually non-isomorphic) graphs on
vertices, and let be the number of distinct decks that can be
constructed from these graphs. Then the difference measures
how many graphs cannot be reconstructed from their decks. In particular, the
graph reconstruction conjecture is true for -vertex graphs if and only if
.
We give a framework based on Kocay's lemma to study this discrepancy. We
prove that if is a matrix of covering numbers of graphs by sequences of
graphs, then . In particular, all
-vertex graphs are reconstructible if one such matrix has rank . To
complement this result, we prove that it is possible to choose a family of
sequences of graphs such that the corresponding matrix of covering numbers
satisfies .Comment: 12 pages, 2 figure
The Graph Structure of Chebyshev Polynomials over Finite Fields and Applications
We completely describe the functional graph associated to iterations of
Chebyshev polynomials over finite fields. Then, we use our structural results
to obtain estimates for the average rho length, average number of connected
components and the expected value for the period and preperiod of iterating
Chebyshev polynomials
Embedding bounded degree spanning trees in random graphs
We prove that if a tree has vertices and maximum degree at most
, then a copy of can almost surely be found in the random graph
.Comment: 14 page
Partitions and Coverings of Trees by Bounded-Degree Subtrees
This paper addresses the following questions for a given tree and integer
: (1) What is the minimum number of degree- subtrees that partition
? (2) What is the minimum number of degree- subtrees that cover
? We answer the first question by providing an explicit formula for the
minimum number of subtrees, and we describe a linear time algorithm that finds
the corresponding partition. For the second question, we present a polynomial
time algorithm that computes a minimum covering. We then establish a tight
bound on the number of subtrees in coverings of trees with given maximum degree
and pathwidth. Our results show that pathwidth is the right parameter to
consider when studying coverings of trees by degree-3 subtrees. We briefly
consider coverings of general graphs by connected subgraphs of bounded degree
The Cover Pebbling Number of Graphs
A pebbling move on a graph consists of taking two pebbles off of one vertex
and placing one pebble on an adjacent vertex. In the traditional pebbling
problem we try to reach a specified vertex of the graph by a sequence of
pebbling moves. In this paper we investigate the case when every vertex of the
graph must end up with at least one pebble after a series of pebbling moves.
The cover pebbling number of a graph is the minimum number of pebbles such that
however the pebbles are initially placed on the vertices of the graph we can
eventually put a pebble on every vertex simultaneously. We find the cover
pebbling numbers of trees and some other graphs. We also consider the more
general problem where (possibly different) given numbers of pebbles are
required for the vertices.Comment: 12 pages. Submitted to Discrete Mathematic
- …