1,689 research outputs found
On the Expansion of Group-Based Lifts
A -lift of an -vertex base graph is a graph on
vertices, where each vertex of is replaced by vertices
and each edge in is replaced by a matching
representing a bijection so that the edges of are of the form
. Lifts have been studied as a means to efficiently
construct expanders. In this work, we study lifts obtained from groups and
group actions. We derive the spectrum of such lifts via the representation
theory principles of the underlying group. Our main results are:
(1) There is a constant such that for every , there
does not exist an abelian -lift of any -vertex -regular base graph
with being almost Ramanujan (nontrivial eigenvalues of the adjacency matrix
at most in magnitude). This can be viewed as an analogue of the
well-known no-expansion result for abelian Cayley graphs.
(2) A uniform random lift in a cyclic group of order of any -vertex
-regular base graph , with the nontrivial eigenvalues of the adjacency
matrix of bounded by in magnitude, has the new nontrivial
eigenvalues also bounded by in magnitude with probability
. In particular, there is a constant such that for
every , there exists a lift of every Ramanujan graph in
a cyclic group of order with being almost Ramanujan. We use this to
design a quasi-polynomial time algorithm to construct almost Ramanujan
expanders deterministically.
The existence of expanding lifts in cyclic groups of order
can be viewed as a lower bound on the order of the largest abelian group
that produces expanding lifts. Our results show that the lower bound matches
the upper bound for (upto in the exponent)
Steinitz Theorems for Orthogonal Polyhedra
We define a simple orthogonal polyhedron to be a three-dimensional polyhedron
with the topology of a sphere in which three mutually-perpendicular edges meet
at each vertex. By analogy to Steinitz's theorem characterizing the graphs of
convex polyhedra, we find graph-theoretic characterizations of three classes of
simple orthogonal polyhedra: corner polyhedra, which can be drawn by isometric
projection in the plane with only one hidden vertex, xyz polyhedra, in which
each axis-parallel line through a vertex contains exactly one other vertex, and
arbitrary simple orthogonal polyhedra. In particular, the graphs of xyz
polyhedra are exactly the bipartite cubic polyhedral graphs, and every
bipartite cubic polyhedral graph with a 4-connected dual graph is the graph of
a corner polyhedron. Based on our characterizations we find efficient
algorithms for constructing orthogonal polyhedra from their graphs.Comment: 48 pages, 31 figure
Some results on the palette index of graphs
Given a proper edge coloring of a graph , we define the palette
of a vertex as the set of all colors appearing
on edges incident with . The palette index of is the
minimum number of distinct palettes occurring in a proper edge coloring of .
In this paper we give various upper and lower bounds on the palette index of
in terms of the vertex degrees of , particularly for the case when
is a bipartite graph with small vertex degrees. Some of our results concern
-biregular graphs; that is, bipartite graphs where all vertices in one
part have degree and all vertices in the other part have degree . We
conjecture that if is -biregular, then , and we prove that this conjecture holds for several families of
-biregular graphs. Additionally, we characterize the graphs whose
palette index equals the number of vertices
Hamilton decompositions of regular expanders: a proof of Kelly's conjecture for large tournaments
A long-standing conjecture of Kelly states that every regular tournament on n
vertices can be decomposed into (n-1)/2 edge-disjoint Hamilton cycles. We prove
this conjecture for large n. In fact, we prove a far more general result, based
on our recent concept of robust expansion and a new method for decomposing
graphs. We show that every sufficiently large regular digraph G on n vertices
whose degree is linear in n and which is a robust outexpander has a
decomposition into edge-disjoint Hamilton cycles. This enables us to obtain
numerous further results, e.g. as a special case we confirm a conjecture of
Erdos on packing Hamilton cycles in random tournaments. As corollaries to the
main result, we also obtain several results on packing Hamilton cycles in
undirected graphs, giving e.g. the best known result on a conjecture of
Nash-Williams. We also apply our result to solve a problem on the domination
ratio of the Asymmetric Travelling Salesman problem, which was raised e.g. by
Glover and Punnen as well as Alon, Gutin and Krivelevich.Comment: new version includes a standalone version of the `robust
decomposition lemma' for application in subsequent paper
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