In graph theory, a branch of mathematics, a radio coloring of an undirected graph is a form of graph coloring in which one assigns positive integer labels to the graphs
such that the labels of adjacent vertices differ by at least two, and the labels of vertices at distance two from each other differ by at least one. Radio coloring was first studied by Griggs & Yeh (1992), under a different name, L(2,1)-labeling.[1][2] It was called radio coloring by Frank Harary because it models the problem of channel assignment in radio broadcasting, while avoiding electromagnetic interference between radio stations that are near each other both in the graph and in their assigned channel frequencies.
The span of a radio coloring is its maximum label, and the radio coloring number of a graph is the smallest possible span of a radio coloring.[1] For instance, the graph consisting of two vertices with a single edge has radio coloring number 3: it has a radio coloring with one vertex labeled 1 and the other labeled 3, but it is not possible for a radio coloring of this graph to use only the labels 1 and 2.
^ abBroersma, Hajo (2005), "A general framework for coloring problems: old results, new results, and open problems" (PDF), Combinatorial geometry and graph theory(PDF), Lecture Notes in Comput. Sci., vol. 3330, Springer, Berlin, pp. 65–79, doi:10.1007/978-3-540-30540-8_7, ISBN 978-3-540-24401-1, MR 2172960. See in particular Section 3, "Radio coloring".
^Griggs, Jerrold R.; Yeh, Roger K. (1992), "Labelling graphs with a condition at distance 2", SIAM Journal on Discrete Mathematics, 5 (4): 586–595, doi:10.1137/0405048, MR 1186826.
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