GROUP TESTING IN BIPARTITE GRAPHS

Abstract

This paper investigates the group testing problem in graphs as follows. Given a graph $G=(V,E)$, determine the minimum number $t(G)$ such that $t(G)$ tests are sufficient to identify an unknown edge $e$ with each test specifies a subset $X\subseteq V$ and answers whether the unknown edge $e$ is in $G[X]$ or not. Damaschke proved that $\lceil\log_2 e(G)\rceil \le t(G) \le \lceil\log_2 e(G)\rceil +1$ for any graph $G,$ where $e(G)$ is the number of edges of $G$. While there are infinitely many complete graphs that attain the upper bound, it was conjectured by Chang and Hwang that the lower bound is attained by all bipartite graphs. This paper verifies the conjecture for bipartite graphs $G$ with $e(G)\le 2^4$ or $2^{k-1} \lt e(G) \le 2^{k-1} + 2^{k-3}+2^{k-6}+19 \cdot 2^{\frac{k-7}{2}}-1$ for $k \ge 5$.