The \(p-\)Arboricity of Outerplanar Graphs

The p-arboricity of a graph G, denoted by \(arb_p(G)\) $ar{b}_p(G)$ , is the smallest number of colors needed to color the edges of G such that every cycle C of G receives at least \(min\{|C|,p+1\}\) $min\{|C|,p+1\}$ colors. In 2019, Bartnicki et al. (Discrete Math., 342:1343–1350, 2019) proposed the following two conjectures: (1) If G is an outerplanar graph, then \(arb_p(G)\le 2p+1\) $ar{b}_p(G)\le 2p+1$ for any integer \(p\ge 2\) $p\ge 2$ . (2) For any integer \(p\ge 2\) $p\ge 2$ , there exists an outerplanar graph G with \(arb_p(G)= 2p+1\) $ar{b}_p(G)=2p+1$ . In this paper we show that the first conjecture does not hold but the second one is true. In addition, we prove that the p-arboricity of an outerplanar graph is at most \(3\cdot 2^{2p+1}-3\) $3·2^{2p+1}-3$ if \(p\ge 3\) $p\ge 3$ . Moreover, there exists an outerplanar graph whose p-arboricity is \(2^{\frac{p+3}{2}}-3\) $2^{\frac{p+3}{2}}-3$ if \(p\ge 3\) $p\ge 3$ is an odd integer.