Geodesic
Edge $k$-$q$-Colorability of Graphs
Journal of Graph Algorithms and Applications, Tome 22 (2018) no. 2, pp. 193-206.

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Given positive integers $k$, $q$, we say that a graph is edge $k$-$q$-colorable if its edges can be colored in such a way that the number of colors incident to each vertex is at most $q$ and that the size of a largest color class is at most $k$. The problem of minimizing $k$ for a given $q$ was considered in [T. Larjomaa and A. Popa, The min-max Edge q-coloring Problem, Journal of Graph Algorithms and Applications, vol 19(1) pp. 505-528 (2015)]. In this paper, we first fix $k=2$ and give an $O(\min\;\{m^2\sqrt{n/\log m}\;,\;nm^{1.5}\})$-time algorithm which given an arbitrary graph $G$ with $n$ vertices and $m$ edges, and a positive integer $q$ decides whether $G$ is $2$-$q$-colorable and outputs a $2$-$q$-coloring if such a coloring exists. Then, we fix $q=2$ and we focus on cubic graphs. In particular, we prove that every cubic graph admits a $4$-$2$-coloring such that the corresponding edge decomposition uses only paths. We give an $O(n\log ^2n)$-time algorithm constructing such a decomposition. 
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     author = {Selma Djelloul and Odile Favaron and Mekkia Kouider},
     title = {Edge $k$-$q${-Colorability} of {Graphs}},
     journal = {Journal of Graph Algorithms and Applications},
     pages = {193--206},
     publisher = {mathdoc},
     volume = {22},
     number = {2},
     year = {2018},
     doi = {10.7155/jgaa.00464},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.7155/jgaa.00464/}
}
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Selma Djelloul; Odile Favaron; Mekkia Kouider. Edge $k$-$q$-Colorability of Graphs. Journal of Graph Algorithms and Applications, Tome 22 (2018) no. 2, pp. 193-206. doi : 10.7155/jgaa.00464. http://geodesic.mathdoc.fr/articles/10.7155/jgaa.00464/

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