Let G\documentclass[12pt]{minimal}
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\begin{document}$$G$$\end{document} be a planar graph with Δ≥8\documentclass[12pt]{minimal}
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\begin{document}$$\varDelta \ge 8$$\end{document} and v\documentclass[12pt]{minimal}
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\begin{document}$$v$$\end{document} be a vertex of G\documentclass[12pt]{minimal}
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\begin{document}$$G$$\end{document}. It is proved that χ′′(G)=Δ+1\documentclass[12pt]{minimal}
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\begin{document}$$\chi ''(G)=\varDelta +1$$\end{document} if v\documentclass[12pt]{minimal}
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\begin{document}$$v$$\end{document} is not incident with chordal 5\documentclass[12pt]{minimal}
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\begin{document}$$5$$\end{document}- or 6\documentclass[12pt]{minimal}
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\begin{document}$$6$$\end{document}-cycles by Shen et al. (Appl Math Lett 22:1369–1373, 2009), or v\documentclass[12pt]{minimal}
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\begin{document}$$v$$\end{document} is not incident with 2\documentclass[12pt]{minimal}
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\begin{document}$$2$$\end{document}-chordal 5\documentclass[12pt]{minimal}
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\begin{document}$$5$$\end{document}-cycle by Chang et al. (Theor Comput Sci 476:16–23, 2013). In this paper we generalize these results and prove that if v\documentclass[12pt]{minimal}
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\begin{document}$$v$$\end{document} is not incident with chordal 6\documentclass[12pt]{minimal}
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\begin{document}$$6$$\end{document}-cycle, or chordal 7\documentclass[12pt]{minimal}
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\begin{document}$$7$$\end{document}-cycle, or 2\documentclass[12pt]{minimal}
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\begin{document}$$2$$\end{document}-chordal 5\documentclass[12pt]{minimal}
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\begin{document}$$5$$\end{document}-cycle, then χ′′(G)=Δ+1\documentclass[12pt]{minimal}
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\begin{document}$$\chi ''(G)=\varDelta +1$$\end{document}.
