an:01396798
Zbl 0931.05032
Borodin, O. V.; Kostochka, A. V.; Raspaud, A.; Sopena, E.
Acyclic coloring of 1-planar graphs
RU
Diskretn. Anal. Issled. Oper., Ser. 1 6, No. 4, 20-35 (1999).
00063037
1999
j
05C15
1-planar graph; acyclic coloring
A graph is said to be 1-planar if it can be embedded into the plane so that each of its edges is crossed by at most one other edge. A coloring of the vertices of a graph is said to be acyclic if every cycle contains at least three colors. The acyclic chromatic number \(a(G)\) of a graph \(G\) is the minimal \(k\) such that \(G\) admits an acyclic \(k\)-coloring. \textit{O.~Borodin} [Discrete Math. 25, No. 3, 211-236 (1979; Zbl 0406.05031)] proved the conjecture of Gr??nbaum that every planar graph is acyclically 5-colorable. The main result of the article under review states that every 1-planar graph can be colored in 20 colors. Note that the largest known value among the acyclic chromatic numbers of 1-planar graphs is 7.
A.Yu.Vesnin (Novosibirsk)
Zbl 0406.05031