A Short Proof of 7-Colorability of $\frac{3}{3}$-Power of Subcubic Graphs

Abstract

For any $k\in {\mathbb {N}}$, the k-subdivision of graph G is a simple graph $G^{\frac{1}{k}}$, which is constructed by replacing each edge of G with a path of length k. In (Iradmusa in Discrete Math 310(10–11):1551–1556, 2010), the mth power of the n-subdivision of G has been introduced as a fractional power of G, denoted by $G^{\frac{m}{n}}$. Wang and Liu (Discrete Math Algorithms Appl 10(3):1850041, 2018) showed that $\chi (G^{\frac{3}{3}})\le 7$ for any subcubic graph G. In this note, a short proof is given for this theorem by use of incidence chromatic number.