Binary Labeling of Graphs |
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Authors: | Louis Caccetta Rui-Zhong Jia |
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Institution: | 1. School of Mathematics and Statistics, Curtin University of Technology, GPO Box U 1987, Perth, 6001, Western Australia
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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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