Total Coloring of Planar Graphs Without Chordal Short Cycles

Abstract

Let $$G$$G be a planar graph with $$\varDelta \ge 8$$Δ?8 and $$v$$v be a vertex of $$G$$G. It is proved that $$\chi ''(G)=\varDelta +1$$???(G)=Δ+1 if $$v$$v is not incident with chordal $$5$$5- or $$6$$6-cycles by Shen et al. (Appl Math Lett 22:1369---1373, 2009), or $$v$$v is not incident with $$2$$2-chordal $$5$$5-cycle by Chang et al. (Theor Comput Sci 476:16---23, 2013). In this paper we generalize these results and prove that if $$v$$v is not incident with chordal $$6$$6-cycle, or chordal $$7$$7-cycle, or $$2$$2-chordal $$5$$5-cycle, then $$\chi ''(G)=\varDelta +1$$???(G)=Δ+1.