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Friday May 11th 

Invited Talk 

Friday 3:00–4:00 109 Oelman Hall 

s-arc transitivity and local s-arc transitivity for graphs 

Cheryl Praeger 

University of Western Australia 

Abstract 

An s-arc of a graph is an ordered path of length s (possibly self-intersecting), and a graph is s-arc 

transitive if all s-arcs are equivalent under automorphisms. Their study was initiated by a famous 

result of Tutte in the late 1940’s. There are many beautiful examples, including most of the polygonal 

graphs of Manley Perkel. My work in the 1990’s showed how to manage the family of s-arc transitive 

graphs (previously thought ‘wild’ or unmanageable because of a result of Babai). The trick was to 

study their normal quotients, identifying basic types and studying those. 

Locally s-arc transitive graphs comprise a larger family in which many have intransitive automorphism 

groups. It was therefore a little surprising to my colleagues and me to discover that the same 

approach to analysis would work for the locally s-arc transitive graphs. I will tell some of this story. 

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Contributed Talks 


Cyclic Arrangements of k-sets with Local Intersection Constraints 

Dan Pritikin*, Tao Jiang, and Manley Perkel 

Miami University of Ohio 

Abstract 

Given positive integers n, k, s with 0 < k < n, does there exist a cyclic ordering of the k-sets of 

{1, 2, . . ., n} such that every s consecutive k-sets are pairwise intersecting? For a given n and k, let 

f(n, k) denote the maximum s for which such an ordering exists. Phrased in terms of graphs, f(n, k) 

is the largest s such that the complement of the Kneser graph K(n, k) contains the s’th power of some 

Hamiltonian cycle in that complement. 

For each n ≥ 6 we show that f(n, 2) = 3. We show that f(n, 3) equals either 2n −8 or 2n −7 when 

n is sufficiently large, conjecturing that 2n − 8 is the correct value. For each k ≥ 4 and n sufficiently 

large we show that 

2n k−2 

(k − 2)! 

(7k 

− 2)nk−3 

2 − 

(k − 3)! + O(nk−4 ) ≤ f(n, k) ≤ 2nk−2 

(k − 2)! 

where c is an absolute positive constant. This is a preliminary report. 


(7k 

− c)nk−3 

2 − 

(k − 3)! 

Non-Cayley vertex-transitive graphs of order p k and valence 2p + 2 

Yuqing Chen 

Wright State University 

Abstract 

We present a family of vertex-transitive graphs of order p k and valence 2p + 2, where p is an odd 

prime and k > 3. The valence of 2p + 2 is the lowest possible for a non-Cayley vertex-transitive graph 

of order a power of p. 

2 

,


Representations of Graphs modulo n: Some Problems 

A. B. Evans*, D. A. Narayan, and J. Urick 


Abstract 

A graph G has a representation modulo n if there exists an injective map f : V (G) → {0, 1, ..., n} 

such that vertices u and v are adjacent if and only if f(u) − f(v) is relatively prime to n. The 

representation number rep(G) is the smallest n such that G has a representation modulo n. We pose 

a collection of related open problems. 


Orthomorphisms of Z3 × Z9 

James Carter 


Abstract 

In order to study orthomorphisms of groups of the form G ∼ = 

n 

i=1 

Z p n i 

i where the pi are not necessarily 

distinct, we begin with a small group Z2 × Z4. All orthomorphisms of this group are explicitly listed 

in Evans (1991) as found by exhaustive computer search. This data was first established by Johsnson, 

Dulmage, and Mendelsohn in 1961. The interesting characteristic of the group in question is its clique 

number, ω, which is the largest value of r for which the orthomorphsim graph admits an r-clique. 

The value of ω is known only for a very few classes of groups, though some bounds are known to 

exist for arbitrary groups. Since the number of orthomorphisms of a group with the structure of G 

undergoes combinatorial explosion very quickly, we first seek a pattern in a small group about which 

much is known, that can be extended to successively larger groups with more generalized structures. 

Therefore, this is an investigation of the group H ∼ = Z3 × Z9, using the homomorphic image under the 

mapping f((h1, h2)) = (h1, h3), h3 ≡ h2 mod 3, a pattern in Z2 × Z4, and MATLAB software. 

3


Non-embeddable Quasi-Derived Designs 

Tariq Alraqad 

Central Michigan University 

Abstract 

A derived design of a symmetric (v, k, λ)–design D, is a 2–(k, λ, λ − 1) design D ′ obtained from D 

by removing a block B and replacing every other block A by A B. A 2-design that has parameters 

of a derived design (equivalently k = λ + 1) is called a quasi-derived design. Moreover, it is said to be 

embeddable if it is the derived design of a symmetric design. 

The main result is a method of constructing a non-embeddable quasi-derived design from a quasiderived 

design and an α-resolvable design. This method is a generalization of techniques used by van 

Lint and Tonchev (1984,1993) and Kageyama and Miao (1996). As applications, several new families 

of non-embeddable quasi-derived designs are constructed. 


Classification of Orthogonal Arrays by Integer Programming 

Dursun Bulutoglu 

Air Force Institute of Technology 

Abstract 

In Bulutoglu and Margot (2007) the problem of classifying all isomorphism classes of OA(N, k, s, t)s 

is shown to be equivalent to finding all isomorphism classes of non-negative integer solutions to a system 

of linear equations under the symmetry group of the system of equations. In this paper a branchand-cut 

algorithm developed by Margot (2002, 2003a, 2003b, 2005) for solving integer programming 

problems with large symmetry groups is used to find all non-isomorphic OA(24, 7, 2, 2)s, OA(24, k, 2, 3)s 

for 6 ≤ k ≤ 11, OA(32, k, 2, 3)s for 6 ≤ k ≤ 11, OA(40, k, 2, 3)s for 6 ≤ k ≤ 10, OA(48, k, 2, 3)s for 6 ≤ 

k ≤ 8, OA(56, k, 2, 3)s, OA(80, k, 2, 4)s, OA(112, k, 2, 4)s, for k = 6, 7, OA(64, k, 2, 4), OA(96, k, 2, 4)s 

for k = 7, 8, and OA(144, k, 2, 4)s for k = 8, 9. Moreover further applications to classifying covering 

arrays with the minimum number of runs and packing arrays with the maximum number of runs are 

presented. I am going to talk about these new developments regarding orthogonal arrays. 

4

Saturday Morning May 12th 


Saturday 9:00–9:25 109 Oelman Hall 

Graph Structures and Excluded Minors 

John Maharry 

The Ohio State University at Marion 

Abstract 

In this talk, I will discuss recent research on various unavoidable minor structures in sufficiently 

large, well-connected graphs. There are several types of these Ramsey-type problems. Many of these 

theorems are based on the ideas of tree-width, near-surface embeddings and vortex structures from 

Robertson and Seymour’s Graph Minors Theory. One nice conjecture in this area is that for any k, 

all sufficiently large 2a + 1-connected graphs must contain a Ka,k-minor. 


The maximum size of an edge cut and graph homomorphisms 

Ju Zhou 

West Virginia University 

Abstract 

A k-cut l-cover of a graph H is a collection F{D1, D2, · · · , Dk} of edge cuts of H such that every 

edge of H lies in exactly l members of F. For a graph G, let b(G) denote the maximum size of an 

edge cut in G. In this paper, it is showed that if G and H are graphs such that H has a k-cut l-cover, 

and if there is a graph homomorphism from G to H, then b(G) ≥ l 

k |E(G)|. When H = Kp/q, a result 

of Erdös in 1979 is generalized and proved to be best possible. When H = Kp:q, a result analogue to 

the result of Erdös in 1979 is got and proved to be best possible. 

5


Nowhere zero 4-flow in regular matroids 

Taoye Zhang 


Abstract 

Let K S2 

5 be the graph obtained from a Petersen graph by contracting 3 edges of a perfect matching. 

We prove that if a coloopless regular matroid M does not have a minor in {M(K S2 

5 ), M ∗ (K5)}, then 

M admits a nowhere-zero 4-flow. 


Some Open Problems Involving Vertex Replacement Rules 

Michelle Previte 

Pennsylvania State University Erie 

Abstract 

Given an initial graph G, one may apply a rule R to G which replaces certain vertices of G with 

other graphs called replacement graphs to obtain a new graph R(G). By iterating this procedure, a 

sequence of graphs {R n (G)} is obtained. When each graph in this sequence is normalized to have 

diameter one, the resulting sequence may converge in the Gromov-Hausdorff metric to a fractal. 

In this talk, vertex replacement rules will be defined and examples will be given of replacement 

rules which yield converging and diverging normalized sequences of graphs. Also, results giving the 

topological and Hausdorff dimensions of the limits of vertex replacements will be presented. We will 

see that it is easy to constuct examples of rules which do and do not yield fractals. Finally, several 

open problems will be discussed. 

6


Bounding the genus of generalized n-gons 

Clifton E. Ealy 

Western Michigan University 

Abstract 

A generalized polygon is a bipartite graph of finite diameter whose girth is twice its diameter. A 

generalized n-gon (for n at least 2) is a generalized polygon of diameter n. Generalized n-gons are 

examples of extremal graphs. In this talk, we will discuss upper and lower bounds for the graph 

theoretic genus of a generalized n-gon. 


Broadcast tree decompositions of Dn 

Matthew Walsh 

Indiana University-Purdue University Fort Wayne 

Abstract 

A broadcast tree is a directed rooted tree with all arcs directed away from the root; such a tree on n 

vertices is minimum if a message from the root can be propagated through the tree in at most ⌈log 2 n⌉ 

“rounds”, where each round consists of some or all of the vertices already in receipt of the message 

transmitting it to uninformed vertices via the arcs of the tree. Let Dn denote the complete directed 

graph, representing a network where any participant can initiate contact with any other participant; 

in this talk I shall demonstrate that the arcs of Dn can be decomposed into minimum broadcast trees 

with distinct roots for all n > 0. Such a decomposition gives a systematic way to broadcast a message 

from any vertex; further, it does not require routing information to be sent with each message. 

7


Recycling Decycling 

Chip Vandell 

Indiana University-Purdue University Fort Wayne 

Abstract 

In this talk we will revisit the parameter ▽(G), the decycling number of a graph and look at how 

it is affected when the additional constraint of independence is put on the decycling set. In particular 

we will look at when decycling sets with this additional constraint exist, then find the constrained 

decycling number for several classes of graphs. 


DNA Codewords and De Bruijn Sequences 

Stephen Hartke 

University of Illinois at Urbana-Champaign 

Abstract 

A De Bruijn sequence is a cyclic string of length n over some alphabet S such that all of the 

substrings of k consecutive letters are distinct under some natural equivalence relation. Given S and 

k, the motivating question is to find a maximum length De Bruijn sequence. Sixty years ago, De 

Bruijn proved that a De Bruijn sequence of length |S| k exists when the equivalence relation is straight 

equality. 

When the alphabet is S = {A, C, G, T }, De Bruijn sequences are useful for the design and testing 

of DNA codewords. Since DNA can twist back on itself, we require the k-substrings to be distinct 

not only under equality but also reverse complementation. Results will be presented on the maximum 

length of a De Bruijn sequence with this equivalence relation. 

8


Infinite families of Steiner 3-designs with block size 6 

Niranjan Balachandran 

The Ohio State University 

Abstract 

We generalize a construction of Hanani which produces a Steiner 3-design on a larger set of points 

starting from a smaller Steiner 3-design, with block size 6. We then construct several infinite families 

of Steiner 3-designs with block size 6. We also indicate the existence of several new Steiner designs 

with block size 5 as a consequence of the so-called product theorem. 


On the role of solvable groups in cage constructions 

Robert Jajcay* and Geoff Exoo 

Indiana State University 

Abstract 

Almost all of the known cages as well as the majority of the current record holders in the unsettled 

cases exhibit a very high level of symmetry. It is unclear, however, whether high symmetry is inherently 

a part of the game or it is simply the case that highly symmetric graphs are easier to handle. In our 

talk, we present results that may seem to suggest certain limitations of the use of groups in cage 

constructions. 

Our main results concern two of the most widely used constructions: Cayley graphs and voltage 

graphs. In each case, we will show how the length of the derived series of a solvable (finite) group 

G used as the underlying group of a Cayley graph or as the group of voltages gives rise to an upper 

bound on the girth of the resulting graph. 

9

Harary Memorial Lecture 


Recent results on graph searching 

Derek Corneil 

University of Toronto 

Abstract 

Searching a graph is a fundamental operation that is the cornerstone of many graph algorithms. 

In the early 1970s most graph searching was either Breadth First Search (BFS) or Depth First Search 

(DFS). Later, Lexicographic BFS (LBFS) was introduced as a simple linear time algorithm to recognize 

chordal graphs. Somewhat surprisingly, LBFS was largely ignored until the 1990s, when it was shown 

to be useful for various problems on restricted families of graphs. In particular, LBFS was applied 

in a multi-sweep fashion where ties of which vertex to choose next were broken by appealing to a 

previous sweep. Correctness proofs of many of these algorithms used an elegant 4-vertex condition that 

characterizes vertex orderings that are LBFS orderings. Since the late 1990s, many new applications of 

LBFS have appeared. Furthermore, similar 4-vertex characterizations have been discovered for other 

graph searches such as BFS, DFS and Lexicographic DFS. 

In this talk, we will give an overview of this work, as well as a summary of open problems. 

10

Conference Honorees 

Terry McKee 

Following his mathematics Ph.D. in 1974 at Wisconsin, Dr. Terry A. McKee was at Western 

Illinois University for two years. The rest of his career has been right here at Wright State, including 

milestones in 1979 (promotion to Associate Professor), 1981 (tenure), and 1985 (promotion to Professor). 

Over the years, Terry’s path of scholarship has taken him from model theory in mathematical 

logic, the field of his doctoral studies and one of the most abstract in all of mathematics, to graph 

theory in discrete mathematics, one of the most readily applicable and accessible branches of our 

subject. His prolific record of research has produced (so far) over 100 published articles and garnered 

nearly a decade of funding from the Office of Naval Research. 

Terry is likewise a smashing success in the classroom. In fact, his outstanding teaching was recognized 

in awards from the College of Science and Mathematics, the President of Wright State University, 

the WSU Alumni Association, and SOCHE - the Southwestern Ohio Council for Higher Education. 

In 1991, Terry garnered the Presidential Award for Outstanding Faculty Member university-wide. 

Ten years later, he was tapped to become Associate Dean, a position Terry held until his midsummer 

retirement. 

Thus, Terry’s career at Wright State has spanned thirty years, throughout which he has been 

active and productive in research. Happily for the department that productivity will continue, and 

will continue here, as Terry has moved back to the department from the CoSM offices and is banging 

away today on his graph theory. 

A telling note about the man appears on the wall of his small, windowless office - not the teaching 

awards, not the certificates recognizing his decades on the faculty, not even his Presidential Award. 

Instead, we see a plaque bearing his name from Dayton’s Community Blood Center, thanking him for 

being a ten-gallon blood donor. 

Manley Perkel 

Dr. Manley Perkel, a native of South Africa, completed his mathematics Ph.D. at Michigan 

in 1977. In a manner of speaking, he then followed in Terry’s footsteps, taking a faculty position at 

Western Illinois and thence coming to Wright State. Following his initial appointment in 1978, Manley 

earned promotion to Associate Professor with tenure in 1985 and promotion to Professor in 1995. 

Exemplary teaching has been a hallmark of Manley’s career from start to finish. Indeed, while he 

was still an Assistant Professor, he won an award for outstanding teaching from what was then WSU’s 

College of Science and Engineering; and also in his very last years he won one of the 2005-2006 College 

of Science and Mathematics Outstanding Teaching Awards. A student in his Spring 2006 Modern 

Algebra class who heard of his forthcoming retirement wrote on an evaluation form, ”Don’t let him 

retire! Give him all the money he wants. Throw in a bigger office and a car, too.” 

Manley’s scholarship, focused on algebra, combinatorics, and graph theory, has taken him far and 

wide over the years. For example, on a six-week visit to Australia and New Zealand, he lectured at 

universities in Perth, Adelaide, Melbourne, Canberra, Sydney, and Auckland - and took a gigantic 

bungee jump along the way, the photographic proof of which resides on his office wall. Moreover, 

11

Manley’s international outlook and expertise have led him to guest lecture three times in Regional 

Studies classes and the like, and in nearly twenty Spanish classes, all just in the past four years. 

Due to the confidence of his colleagues, Manley has filled key leadership roles throughout his tenure 

at WSU. As a small sample, witness his repeated election to the department’s Steering Committee, the 

university Academic Council, and the university Senate. Among other accomplishments, his creativity 

and ingenuity were critical for the university’s very creation of both the faculty rank of lecturer and 

the university Senate. Notably, Manley served a five-year term as department chair, having been 

appointment to a three-year term - all he wanted - and having been asked for two one-year encores by 

the dean. 

Manley’s activities during his retirement so far have included a special visiting appointment this 

past fall semester at Miami University where he taught and pursued some new research with his 

collaborators there. Beyond that, Manley tells us that his plans are less clear, though they will surely 

feature the international travel of which he is so fond, and may also include more teaching or research 

or both. We certainly hope at least some of that will occur at WSU! 

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Saturday Afternoon May 12th 



Graceful Labelings of Cycles 

Jay Bagga*, Adrian Heinz, and M. Mahbubul Majumder 

Ball State University 

Abstract 

Rosa (1967) showed that a cycle Cn has a graceful labeling if and only if n ≡ 0(mod 4) or n ≡ 

3(mod 4). In this paper we investigate several properties of graceful labelings of such cycles. We also 

present an algorithm for finding all graceful labelings of cycles, and discuss its implementation. 


An O(n 3 )-time recognition algorithm for hhds-free graphs 

Elaine Eschen 


Abstract 

The class of hhds-free graphs properly generalizes the classes of strongly chordal graphs and 

distance-hereditary graphs, and forms a restriction of the class of perfectly orderable graphs. The 

problem of recognizing hhds-free graphs in polynomial time was posed by Brandstdt (Problem session: 

Dagstuhl Seminar No. 04221, 2004), and Nikolopoulos and Palios (Proceedings of the 31st International 

Workshop on Graph Theoretic Concepts in Computer Science, WG2005) gave an O(mn 2 ) 

algorithm. We present an O(n 3 )-time algorithm for the problem. Our algorithm demonstrates a relationship 

between hhds-free graphs and strongly chordal graphs similar to that which is known to exist 

between hhd-free graphs and chordal graphs. This is joint work with Chính Hoàng (Wilfrid Laurier 

University) and R. Sritharan (The University of Dayton). 

13


Results on Generalized Powers of Graphs 

Yang Xiang 

Kent State University 

Abstract 

We will present our results on ”Generalized Powers of Graphs and Their Algorithmic Use”. We will 

introduce two new types of coloring of graphs, which generalize the well-known Distance-k-Coloring. 

The motivation comes from the frequency assignment problem in heterogeneous multihop radio networks, 

where different radio stations may have different transmission ranges. Let G = (V, E) be a 

graph modelling a radio network, and assume that each vertex v of G has its own transmission radius 

r(v), a nonnegative integer. We define r-coloring (r + -coloring) of G as an assignment Φ : V ↦→ 0, 1, 2, ... 

of colors to vertices such that Φ(u) = Φ(v) implies dG(u, v) > r(v) + r(u) (dG(u, v) > r(v) + r(u) + 1, 

respectively). The r-Coloring problem (the r + -Coloring problem) asks for a given graph G and a 

radius-function r : V ↦→ N ∪0, to find an r-coloring (an r + - coloring, respectively) of G with minimum 

number of colors. Using a new notion of generalized powers of graphs, we investigate the complexity 

of the r-Coloring and r + -Coloring problems on several families of graphs, such as k-chordal graphs, 

weakly chordal graphs, AT-free graphs, co-comparability graphs, interval graphs, circular arc graphs 

and strongly chordal graphs. Most of these results were obtained in collaboration with Andreas 

Brandstädt, Feodor Dragan and Chenyu Yan. 


Collective tree spanners of special classes of graphs 

Chenyu Yan 

Kent State University 

Abstract 

We say that a graph G = (V, E) admits a system of µ collective additive tree r-spanners if there is 

a system T (G) of at most µ spanning trees of G such that for any two vertices x, y of G a spanning tree 

T ∈ T (G) exits such that dT(x, y) ≤ dG(x, y)+r. In this talk, we give a survey of the recent results on 

collective tree spanners problem. In particular, we will present the results of collective tree spanners on 

chordal, AT-free related, homogeneously orderable graphs and graphs with bounded chrodality, tree 

width and clique-width. Among other results we show that 

• Every chordal graph admits a system of O(log n) collective additive tree 2-spanners. 

• Every AT-free graphs admits a system of 2 collective additive tree 2-spanners. 

• Every homogeneously orderable graph admits a system of O(logn) collective additive tree 2spanners. 

• Every k-chordal graph admits a system of O(log n) collective additive tree ⌊(k +1)/3⌋-spanners. 

All the above collective tree spanners can be constructed in polynomial time. This is joint work with 

Derek G. Corneil and Feodor F. Dragan. 

14


Logarithmic Coloring: How Hard is it to Determine Isomorphism? 

Christopher Augeri*, Barry Mullins, Leemon Baird, Dursun Bulutoglu, and Rusty 

Baldwin 

Air Force Institute of Technology 

Abstract 

We are interested in canonically ordering multi-dimensional data such as node positions of an 

unmanned aerial vehicle swarm. If we assume the data is represented by a pair-wise distance matrix, 

this problem is equivalent to determining graph isomorphism. Our approach to finding a canonical 

ordering is based on the PageRank algorithm, which sorts vertices, such as web pages, on the leading 

eigenvector of a perturbed adjacency matrix. The algorithm we present, IsoCanon, orders vertices by 

lexicographically sorting on the inverse of an adjacency matrix. 

Since the inverse of an adjacency matrix may not exist, we use two isomorphism-preserving perturbations 

that apply the Gershgorin Circle theorem to guarantee the inverse exists and so it may 

be efficiently obtained. To increase IsoCanon’s ability to find a canonical ordering, we also apply the 

necessary condition that vertices in the same orbit share the same inverse entries. We have demonstrated 

this algorithm terminates within a logarithmic number of iterations relative to the vertex set 

magnitude and finds a canonical isomorph of many random and regular graphs. 

We first explore how much additional information is needed to enable IsoCanon to canonically label 

all graphs, such as those based on Hadamard matrices. By randomly coloring a logarithmic number 

of vertices prior to executing the algorithm, sufficient noise is introduced to successfully identify all 

tested graphs. We then explore a deterministic method for coloring a logarithmic set of vertices using 

the matrix inverse prior to execution. This current version identifies more difficult graphs, such as 

Paley graphs and many posed by Mathon. The end result is the problem of determining isomorphism 

can be framed as ”how hard is it to make a graph easy to identify?”. 

IsoCanon also has three key features useful for sorting multi-dimensional data: it terminates in 

polynomial time, it canonically orders many graphs, and this ordering is akin to PageRank with respect 

to connectivity and vertex significance. IsoCanon uses parallel numerical libraries to obtain the inverse 

and has been extensively tested on dense graphs having up to 4,000 vertices. 


Linda Lesniak*, R. J. Faudree, and Ingo Schiermeyer 

Drew University 

Abstract 

Let G be a graph of order n and circumference c(G). Let G be the complement of G. We prove 

that max{c(G), c(G)} ≥ ⌈2n ⌉ and show sharpness of this bound. 

3 

15


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. 


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. 

16


Generalizing a theorem of Kundu 

Art Busch 

University of Dayton 

Abstract 

Kundu’s Theorem states that for a graphic sequence D = (d1, d2, . . .,dn) such that (d1 − k, d2 − 

k, . . .,dn − k) is also graphic, there exists a graph G with degree sequence D that contains a k-factor. 

In this talk we interpret this result as a problem on edge-colored complete graphs, and discuss two 

new generalizations. 


On s-hamiltonian line graphs 

Yehong Shao 

Ohio University Southern 

Abstract 

For an integer s ≥ 0, a graph G is s-hamiltonian if for any vertex subset S ⊆ V (G) with |S| ≤ s, 

G − S is hamiltonian. It is well known that if a graph G is s-hamiltonian, then G must be (s + 2)connected. 

The converse is not true, as there exist arbitrarily highly connected non-hamiltonian 

graphs. But for line graphs, we prove that when s ≥ 5, a line graph is s-hamiltonian if and only if it 

is (s + 2)-connected. 

17

Additional Abstracts 


2-arc transitive polygonal graphs of high girth 

Ákos Seress 

The Ohio State University 

Abstract 

A near-polygonal graph is a graph Γ with a distinguished set C of cycles of common length m such 

that each path of length two lies in a unique element of C. If m is the girth of Γ then the graph is 

called polygonal. 

Polygonal graphs are Manley Perkel’s invention. In his thesis, he introduced two infinite families 

of 2-arc transitive, near-polygonal graphs of valency 5 and 6, which are orbital graphs of special linear 

groups. In this talk, we report on computer searches for polygonal graphs in Perkel’s families. The 

largest examples we have found have girth 23 and the number of vertices is over 10 17 . 

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