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Let $G$ be a connected graph of order $n\ge 3$ and let $cE\left(G\right)\to \{1,2,...,k\}$ be a coloring of the edges of $G$ (where adjacent edges may be colored the same). For each vertex $v$ of $G$, the color code of $v$ with respect to $c$ is the $k$-tuple $c\left(v\right)=(a_1,a_2,\cdots ,a_k)$, where $a_i$ is the number of edges incident with $v$ that are colored $i$ ($1\le i\le k$). The coloring $c$ is detectable if distinct vertices have distinct color codes. The detection number $det\left(G\right)$ of $G$ is the minimum positive integer $k$ for which $G$ has a detectable $k$-coloring. We establish a formula for the detection...

Let G be a connected graph of size at least 2 and c :E(G)→{0, 1, . . . , k− 1} an edge coloring (or labeling) of G using k labels, where adjacent edges may be assigned the same label. For each vertex v of G, the color code of v with respect to c is the k-vector code(v) = (a0, a1, . . . , ak−1), where ai is the number of edges incident with v that are labeled i for 0 ≤ i ≤ k − 1. The labeling c is called a detectable labeling if distinct vertices in G have distinct color codes. The value val(c) of...