Conflict-free incidence coloring and two-way radio networks
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In this paper, we introduce the conflict-free incidence coloring of graphs to model a problem of designing two-way radio networks efficiently and economically. Specifically, we call the vertex-edge pair (v,e) an incidence of a graph. A conflict-free incidence coloring of a graph is a coloring of the incidences in such a way that two incidences (u,e) and (v,f) get distinct colors if and only if they conflicts each other, i.e.,(i) u=v, (ii) uv is e or f, or (iii) there is a vertex w such that uw=e and vw=f. The minimum number of colors used among all conflict-free incidence colorings of a graph is the conflict-free incidence chromatic number. For a simple graph with maximum degree Δ, we claim that its conflict-free incidence chromatic number is either 2Δ, 2Δ+1, or 2Δ+2, and each of them can be attained by infinite many graphs. We also show that the conflict-free incidence chromatic number of an outer-1-planar graph with maximum degree Δ is either 2Δ or 2Δ+1, and moreover, characterize all outer-1-planar graphs whose conflict-free incidence chromatic numbers are exactly 2Δ or 2Δ+1.
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