an:06707238
Zbl 1359.05099
Choi, Hojin; Choi, Ilkyoo; Jeong, Jisu; Suh, Geewon
\((1,k)\)-coloring of graphs with girth at least five on a surface
EN
J. Graph Theory 84, No. 4, 521-535 (2017).
00364670
2017
j
05C70 05C07 05C15
improper coloring; discharging; graphs on surfaces
Summary: A graph is \((d_1\ldots,d_r)\)-colorable if its vertex set can be partitioned into \(r\) sets \(V_1,\ldots,V_r\) so that the maximum degree of the graph induced by \(V_i\) is at most \(d_i\) for each \(i\in\{1,\ldots,r\}\). For a given pair \((g,d_1)\), the question of determining the minimum \(d_2=d_2(g,d_1)\) such that planar graphs with girth at least \(g\) are \((d_1,d_2)\)-colorable has attracted much interest. The finiteness of \(d_2(g,d_1)\) was known for all cases except when \((g,d_1)=(5,1)\). \textit{M. Montassier} and \textit{P. Ochem} [Electron. J. Comb. 22, No. 1, Research Paper P1.57, 13 p. (2015; Zbl 1308.05052)] explicitly asked if \(d_2(5,1)\) is finite. We answer this question in the affirmative with \(d_2(5,1)\leq10\); namely, we prove that all planar graphs with girth at least five are \((1,10)\)-colorable. Moreover, our proof extends to the statement that for any surface \(S\) of Euler genus \(\gamma\), there exists a \(K=K(\gamma)\) where graphs with girth at least five that are embeddable on \(S\) are \((1,K)\)-colorable. On the other hand, there is no finite \(k\) where planar graphs (and thus embeddable on any surface) with girth at least five are \((0,k)\)-colorable.
Zbl 1308.05052