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

Pulsed plasma deposition of Fe-C-Cr-W coating on high-Cr-cast iron: Effect of layered morphology and heat treatment on the microstructure and hardness

Pulsed plasma treatment was applied for surface modification and laminated coating deposition on 14.5 wt%-Cr cast iron. The scopes of the research were: (a) to obtain a microstructure gradient, (b) to study the relationship between cathode material and coating layer microstructure/hardness, and (c) to improve coating quality by applying post-deposition heat treatment. An electrothermal axial plasma accelerator with a gas-dynamic working regime was used as plasma source (4.0 kV, 10 kA). The layered structure was obtained by alternation of the cathode material (T1 - 18 wt% W high speed steel and 28 wt% Cr-cast iron). It was found that pulsed plasma treatment led to substrate sub-surface modification by the formation of an 11–18 μm thick remelted layer with very fine carbide particles that provided a smooth transition from the substrate into the coating (80–120 μm thick). The as-deposited coating of 500–655 HV0.05 hardness consisted of “martensite/austenite” layers which alternated with heat-affected layers (layers the microstructure of which was affected by the subsequent plasma pulses). Post-deposition heat treatment (isothermal holding at 950 °C for 2 h followed by oil quenching) resulted in precipitation of carbides M7C3, M3C2, M3C (in Cr-rich layers) and M6C, M2C (in W-rich layers). These carbides were found to be Cr/W depleted in favor of Fe. The carbide precipitation led to a substantial increase in the coating hardness to 1240–1445 HV0.05. The volume fraction of carbides in the coating notably increased relatively to the electrode materials