Injective chromatic index of sparse graphs (Q6145804)

A \(k\)-injective-edge coloring of a graph \(G\) is an assignment of colors, i.e. integers in \(\{1, 2, \ldots , k\}\), to the edges of \(G\) such that \(e_1\) and \(e_3\) receive distinct colors for any three consecutive edges \(e_1\), \(e_2\), \(e_3\) of a path or a 3-cycle. The smallest integer \(k\) such that \(G\) has an injective-edge coloring is called the injective chromatic index of \(G\), denoted by \(\chi^\prime_i(G)\). For a graph \(G\) and a positive integer \(k\), \textit{D. M. Cardoso} et al. [``Injective edge chromatic index of a graph'', Preprint, \url{arXiv:1510.02626}] proved that it is NP-hard to check whether \(\chi^\prime_i(G)=k\). For a simple graph \(G\), that is not a cycle with maximum degree \(\Delta\), \(\chi^\prime_i(G)\le 2(\Delta-1)^2\). \textit{Z. Miao} et al. [Discrete Appl. Math. 310, 65--74 (2022; Zbl 1482.05118)] proposed the following conjecture: Let \(G\) be a simple graph with maximum degree \(\Delta\). Then \(\chi^\prime_i(G)\le \Delta(\Delta-1)\). Miao et al. [loc. cit.] attacked this conjecture using the maximum average degree \(\mathrm{mad}(G)\) of a graph \(G\). For a graph \(G\), its maximum average degree mad\((G)\) is mad\((G) =\max_{\emptyset \ne H\subseteq G}\frac{ 2|E(H)|}{|V(H)|}\). Namely, Miao et al. [loc. cit.] analyzed graphs with maximum degree \(\Delta=4\). They showed that for the graphs \(G\) with \(\Delta (G) = 4\), \(\chi^\prime_i(G)\le 9\) \((\text{resp.}, 10, 11, 12)\) if mad\( (G) < \frac{14} 5\) \((\text{resp.}, 3, \frac{19} 6 ,\frac{ 36}{ 11} )\). Similar results were obtained for the graph with maximum degree 5. \textit{J. Zhu} [Discrete Appl. Math. 334, 119--126 (2023; Zbl 1512.05150)) proved that \(\chi^\prime_i(G)\le 8\) \((\text{resp.}, 9, 10, 11, 13, 14)\) if mad\((G) <\frac{7}{3}\) \((\text{resp.}, \frac{12} 5 ,\frac{ 5} 2 ,\frac{ 18} 7 , \frac{14} 5 , 3)\) for any graphs \(G\) with \(\Delta (G) = 5\). In this paper, the authors improve the upper bounds on the injective chromatic index of such graphs (graphs with maximum degree 5). They show the following: Let \(G\) be a simple and finite graph with \(\Delta(G)=5\). Then \(\chi^\prime_i(G)\le 10\) \((\text{resp.} 11, 12, 13)\) if mad\( (G) <\frac{ 37}{ 14}\) \((\text{resp.} \frac{39}{14},\frac{17}{6}, 3)\).