Pi Mu Epsilon - Mathematical Association of America

Thursday MAA Session #8 August 2, 2012
3:00–3:15
On d-modular Labelings of the Union of Two Cycles
Joseph Buchanan
Illinois State University
For positive integers r and s, let K r×s denote the complete multipartite graph with r parts of
size s each. Let G be a graph with n edges, d be a positive integer such that d|2n and set
c = 2n/d + 1. A d-modular ρ-labeling of G is a one-to-one function f : V (G) → [0, cd) such that
{min(|f(u) − f(v)|, cd − |f(u) − f(v)|): {u, v} ∈ E(G)} = {1, 2, . . . , ⌊ cd ⌋} \ cZ.
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It is known that if a z-partite graph G admits a d-modular ρ-labeling, then there exists a cyclic G-
decomposition of K c×td for every positive integer t such that gcd(t, (z − 1)!) = 1. We investigate
d-modular labelings of the union of two vertex-disjoint cycles.
3:20–3:35
On 2-fold Graceful Labelings
Ellen Sparks
Illinois State University
Let Z denote the set of integers and N denote the set of nonnegative integers. For integers a and
b with a ≤ b, let [a, b] = {x ∈ Z : a ≤ x ≤ b}. For a positive integer k, let 2 K k denote the
2-fold complete mutigraph of order k. Similarly, let 2 [a, b] denote the multiset that contains every
element of [a, b] exactly two times. Let G be a multigraph of size n, order at most n + 1, and edge
multiplicity at most 2. A labeling of G is a one-to-one function f : V (G) → N. If f is a labeling
of G and e = {u, v} ∈ E(G), let ¯f(e) = |f(u) − f(v)|. A 2-fold graceful labeling of G is a
one-to-one function f : V (G) → [0, n] such that { ¯f(e): e ∈ E(G)} is either the multiset 2 [1, n] if 2
n is even or the multiset 2 [1, n−1]
∪ { n+1 } if n is odd.
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A graph G is said to be 2-fold graceful if it admits a 2-fold graceful labeling. It can be shown
that if G with n edges is 2-fold graceful, then there exists a cyclic G-decomposition of 2 K n+1 . El-
Zanati has conjectured that every tree is 2-fold graceful. We investigate 2-fold graceful labelings
of various classes of graphs including several classes of trees.
3:40–3:55
Direct Sums for Graph Automorphisms
Michael Carlyle
Augustana College
A graph automorphism is a mapping from and to itself that preserves vertex adjacency. After a
brief recap of fundamental ideas such as groups and graphs, we will present examples of graphs
whose automorphism groups are “easy” to see, and then we will show graphs whose automorphism
groups are direct sums. We will also show examples where the techniques of direct sums are not
quite enough.
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