On Incidence Coloring of Complete Multipartite and Semicubic Bipartite Graphs

In the paper, we show that the incidence chromatic number $ \chi_i $ of a complete $k$-partite graph is at most $ \Delta + 2 $ (i.e., proving the incidence coloring conjecture for these graphs) and it is equal to $ \Delta + 1 $ if and only if the smallest part has only one vertex (i.e., $ \Delta = n − 1 $). Formally, for a complete k-partite graph $ G = K_{r_1,r_2,...,r_k} $ with the size of the smallest part equal to $ r_1 \ge 1 $ we have $$ \chi_i (G)= \begin{cases} \Delta(G)+1 & \text { if } r_1=1, \\ \Delta(G)+2 & \text { if } r_1>1. \end{cases} $$ In the paper we prove that the incidence 4-coloring problem for semicubic bipartite graphs is \( \mathcal{NP} \)-complete, thus we prove also the \( \mathcal{NP} \)-completeness of L(1, 1)-labeling problem for semicubic bipartite graphs. Moreover, we observe that the incidence 4-coloring problem is \( \mathcal{NP} \)-complete for cubic graphs, which was proved in the paper [12] (in terms of generalized dominating sets).