On the Computational Complexity of Local and Global Covering Numbers

The global and local $\mathcal{G}$-covering number $c_{\mathrm{g}}^{\mathcal{G}}(H)$ and $c_{\mathrm{l}}^{\mathcal{G}}(H)$ encode how well the edges of a graph $H$ can be covered with graphs from a graph class $\mathcal{G}$: in the global setting, we minimize the number of graphs from $\mathcal{G}$ required, in the local setting how often a vertex is hit by the graphs of the cover. Within this work we consider for $\mathcal{G}$ the graph classes $\mathcal{B}$ of all bipartite and $\mathcal{B}_{\mathrm{c}}$ of all complete bipartite graphs. We give a tight lower bound on $c_{\mathrm{l}}^{\mathcal{B}}(H)$ in terms of the fractional chromatic number of $H$, thereby giving a local analogue of a result by Harary, Hsu and Miller. Answering a question by Fishburn and Hammer, we show that it is NP-hard to determine $c_{\mathrm{l}}^{\mathcal{B}_{\mathrm{c}}}(H)$. Further, we provide a finite and monotone graph class $\mathcal{G}$ such that $c_{\mathrm{g}}^{\mathcal{G}}(H)$ can be computed in constant time for every graph $H$ while determining $c_{\mathrm{l}}^{\mathcal{G}}(H)$ is NP-hard. This yields a natural example to a question raised by Knauer and Ueckerdt.