Conjectures

$Lemma$ 2.8 and $Theorem$ $2.11$ might be used to show the following.

Conjecture 3.15   $\rho (K_{n,m})=m$ when $3n-3>m \ge n \ge 3$.

The following would be an extension to $Theorem$ 2.11.

Conjecture 3.16   If $K_{n_1,n_2,...,n_m}$ with $n_1>n_2>...>n_m$ is not a MSG, then          $(m-1)n_m \le \rho (K_{n_1,n_2,...,n_m}) \le (m-1)n_1$.

The following shows good promise and would be an extension of $Lemma$ 2.8.

Conjecture 3.17   If $K_{n_1,n_2,...,n_m}$ with $n_1>n_2>...>n_m$ is a MSG, then $n_i \ge 2n_{i+1} + \sum^m_{j=i+2} n_j$.

The following would be extension to work by [#!hs2!#] and [#!mrss!#].

Conjecture 3.18   $\rho(W_n) =3$ when for $n>4$ and n even.

Conjecture 3.19   $\rho(W_n) =n$ when for $n>4$ and n odd.

The following conjectures appear to be true, but seem difficult to prove or disprove.

Conjecture 3.20   Let $G=(V,E)$ be a graph with $\vert V \vert =n$, then $\rho(G) \le n$.

Conjecture 3.21   Let $G=(V,E)$ be a graph, then finding the mod sum number is np-complete.

Conjecture 3.22   Let $G=(V,E)$ be a graph. There exist $L$ based on $n$ such that if G is not MSG modulo $\cal Z$ $\le L$ then G is not a MSG.

BIBLIOGRAPHY

VITA Christopher D. Wallace
111 Brown Road #26
Johnson City, TN 37615
Phone: (423)952-0545

PROFESSIONAL EXPERIENCE:
Math Tutor, ETSU, Johnson City, TN, 1/95-8/98
Math Lab Coordinator, ETSU, Johnson City, TN, 1/98-8/98
Math Instructor, ETSU, Johnson City, TN, 8/98-present
EDUCATION
B.S. with majors in Computer Science and Math, and concentration in Physics, ETSU, 5/97
M.S. in Mathematical Science, ETSU, 5/99
Graduate Student in Computer Science, ETSU, 8/97-present
PRESENTATIONS:
Progress on Mod Sum Graphs, at the 936$^{th}$ AMS Meeting, Wake Forest University, October 9-10, 1998
HONORS AND AWARDS
Hall of Fame in Computer Science, ETSU, 1994-1997
Outstanding Mathematics Senior, 1996, ETSU
Outstanding Mathematics Senior, 1997, ETSU
Runner Up for Outstanding Undergraduate in Computer Science, 1997, ETSU
Graduated 1997, ETSU
Graduating 1999, ETSU
Member of Gamma Beta, Kappa Mu Epsilon, Phi Kappa Phi, and Upsilon Pi Upsilon Honor Societies