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Binary Labeling of Graphs

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Abstract

Let G = (V, E) be a graph. A mapping f: E(G) → {0, l}m is called a mod 2 coding of G, if the induced mapping g: V(G) → {0, l}m, defined as \(g(v) = \sum\limits_{u \in V,uv \in E} {f(uv)}\), assigns different vectors to the vertices of G. Note that all summations are mod 2. Let m(G) be the smallest number m for which a mod 2 coding of G is possible. Trivially, m(G) ≥ ⌈Log2 |V|⌉. Recently, Aigner and Triesch proved that m(G) ≤ ⌈Log2 |V|⌉ + 4. In this paper, we determine m(G). More specifically, we prove that if each component of G has at least three vertices, then

$$mG = \left\{ {\begin{array}{*{20}c} {k,} & {if \left| V \right| \ne 2^k - 2} \\ {k + 1,} & {else} \\ \end{array} ,} \right.$$

where k = ⌈Log2 |V|⌉.

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Correspondence to Louis Caccetta.

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Caccetta, L., Jia, RZ. Binary Labeling of Graphs. Graphs and Combinatorics 13, 119–137 (1997). https://doi.org/10.1007/BF03352990

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  • DOI: https://doi.org/10.1007/BF03352990

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