Testing Equality in Communication Graphs

Let $G=(V,E)$ be a connected undirected graph with $k$ vertices. Suppose that on each vertex of the graph there is a player having an $n$-bit string. Each player is allowed to communicate with its neighbors according to an agreed communication protocol, and the players must decide, deterministically, if their inputs are all equal. What is the minimum possible total number of bits transmitted in a protocol solving this problem ? We determine this minimum up to a lower order additive term in many cases (but not for all graphs). In particular, we show that it is $kn/2+o(n)$ for any Hamiltonian $k$-vertex graph, and that for any $2$-edge connected graph with $m$ edges containing no two adjacent vertices of degree exceeding $2$ it is $mn/2+o(n)$. The proofs combine graph theoretic ideas with tools from additive number theory.