Group testing in graphs

Abstract

This paper studies 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⊆V and answers whether the unknown edge e is in G[X] or not. Damaschke proved that ⌈log ₂ e(G)⌉≤t(G)≤⌈log ₂ e(G)⌉+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. Later, they proved that the conjecture is true for complete bipartite graphs. Chang and Juan verified the conjecture for bipartite graphs G with e(G)≤2⁴ or $2^{k-1}<e(G)\le 2^{k-1}+2^{k-3}+2^{k-6}+19\cdot 2^{\frac{k-7}{2}}$ for k≥5. This paper proves the conjecture for bipartite graphs G with e(G)≤2⁵ or $2^{k-1}<e(G)\le 2^{k-1}+2^{k-3}+2^{k-4}+2^{k-5}+2^{k-6}+2^{k-7}+27\cdot 2^{\frac{k-8}{2}}-1$ for k≥6.