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<p>For a planar graph <i>G</i> and two distinct vertices <i>u, v</i> ∈ <i>V</i>(<i>G</i>) which share a common neighbor, a coloring is called injective if <i>u</i> and <i>v</i> receive different colors. The injective choosability number of <i>G</i> is the smallest integer <i>k</i> such that <i>G</i> is injective <i>k</i>-choosable, denoted by <i>χ</i><Stack> <sub><i>i</i></sub> <sup><i>l</i></sup> </Stack>(<i>G</i>) In this paper, we prove that <i>χ</i><Stack> <sub><i>i</i></sub> <sup><i>l</i></sup> </Stack>(<i>G</i>) ≤ Δ + 4 if Δ ≥ 15 for <i>G</i> without 3-cycles and disjoint 4-cycles, which improves the results of Li et al. (2022) who gave that <i>χ</i><Stack> <sub><i>i</i></sub> <sup><i>l</i></sup> </Stack>(<i>G</i>) ≤ Δ + 4 if Δ ≥ 22 and <i>χ</i><Stack> <sub><i>i</i></sub> <sup><i>l</i></sup> </Stack>(<i>G</i>) ≤ Δ + 5 if Δ ≥ 15 for <i>G</i> without 3-cycles and disjoint 4-cycles.</p>

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List Injective (Δ + 4)-coloring of Planar Graphs

  • 期刊论文

  • 发表日期：

    2025年12月8日

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  • Yue-hua Bu,
  • Hong-rui Zheng,
  • Hong-guo Zhu

摘要

For a planar graph G and two distinct vertices u, v ∈ V(G) which share a common neighbor, a coloring is called injective if u and v receive different colors. The injective choosability number of G is the smallest integer k such that G is injective k-choosable, denoted by χ i l (G) In this paper, we prove that χ i l (G) ≤ Δ + 4 if Δ ≥ 15 for G without 3-cycles and disjoint 4-cycles, which improves the results of Li et al. (2022) who gave that χ i l (G) ≤ Δ + 4 if Δ ≥ 22 and χ i l (G) ≤ Δ + 5 if Δ ≥ 15 for G without 3-cycles and disjoint 4-cycles.