Sum List Edge Colorings of Graphs

Let $ G = (V,E) $ be a simple graph and for every edge $ \mathcal{e} \in E $ let $ L(e) $ be a set (list) of available colors. The graph $ G $ is called $L$-edge colorable if there is a proper edge coloring $ c $ of $ G $ with $ c(\mathcal{e} ) \in L( \mathcal{e} ) $ for all $ \mathcal{e} \in E $. A function $ f : E \rightarrow \mathbb{N} $ is called an edge choice function of $G$ and $G$ is said to be $f$-edge choosable if $G$ is $L$-edge colorable for every list assignment $L$ with $ |L( \mathcal{e} )| = f( \mathcal{e} ) $ for all $ \mathcal{e} \in E $. Set $ \text{size}(f) = \Sigma_{ \mathcal{e} \in E } f(e) $ and define the sum choice index $ \chi_{sc}^' (G) $ as the minimum of $ \text{size} (f) $ over all edge choice functions $f$ of $G$. There exists a greedy coloring of the edges of $G$ which leads to the upper bound $ \chi_{sc}^′ (G) \le 1/2 \Sigma_{ v \in V } d(v)^2 $. A graph is called sec-greedy if its sum choice index equals this upper bound. We present some general results on the sum choice index of graphs including a lower bound and we determine this index for several classes of graphs. Moreover, we present classes of sec-greedy graphs as well as all such graphs of order at most 5.