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Saturday 3:25–3:50 112 Oelman Hall
Antimagic labelings of regular bipartite graphs:
An application of the Marriage Theorem
Daniel Cranston
DIMACS, Rutgers University
Abstract
A labeling of a graph is a bijective function onto its edges from the set {1, 2, . . ., |E(G)|}. A labeling
is antimagic if for every pair of distinct vertices u and v, the sum of the labels on edges incident to
u is different from the sum of the labels on edges incident to v. We say a graph is antimagic if it
has an antimagic labeling. In 1990, Ringel conjectured that every connected graph other than K2 is
antimagic. The most significant progress was been made by Alon et al. (in 2004), who showed there
exists a constant C, such that if an n-vertex graph G has δ(G) ≥ Cn, then G is antimagic. In this
paper, we show that every regular bipartite graph (with degree at least 2) is antimagic. Our technique
relies heavily on the Marriage Theorem.
Saturday 3:50–4:15 112 Oelman Hall
The degree-associated reconstruction number of a graph
Michael D. Barrus
University of Illinois at Urbana-Champaign
Abstract
Given a graph G, a card of G is an induced subgraph formed by deleting one vertex. The multiset
of cards of G is the deck of G. The well-known Reconstruction Conjecture states that any two graphs
on at least three vertices with the same decks are isomorphic; that is, the deck determines the graph.
One can then define the reconstruction number rn(G) to be the minimum number of cards from the
deck that suffice to determine G. We consider a variation proposed by S. Ramachandran, in which
each card is presented along with the degree of the deleted vertex, yielding a “dacard.” The degreeassociated
reconstruction number drn(G) is the minimum number of dacards that determine G. We
describe several results on drn(G) in the cases where G is a regular graph or a tree. This is joint work
with Douglas B. West.
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