Wegner’s conjecture on distance-two colorings of planar graphs
Establish that for every planar graph G with maximum degree Δ, the distance-2 chromatic number χ2(G) satisfies the bounds 7 if Δ=3, Δ+5 if 4 ≤ Δ ≤ 7, and ⌊(3Δ)/2⌋+1 if Δ ≥ 8.
References
By doing this, they made progress on Wegner's well-known conjecture about the distance 2-chromatic number $\chi_2(G)$ of planar graphs $G$ with maximum degree $\Delta$ (i.e., the classical chromatic number of the graph obtained by adding an edge between each pair of vertices at distance 2 in $G$): \begin{conj}[] Let $G$ be a planar graph with maximum degree $\Delta$. Then, we have: \begin{align*} \chi_2(G) \leq \begin{cases} 7& \textrm{if } \Delta=3,\ \Delta+5& \textrm{if } 4 \leq \Delta \leq 7,\ \lfloor \frac{3\Delta}{2} \rfloor +1 & \textrm{if } \Delta \geq 8.\ \end{cases} \end{align*} \end{conj}