Browsing by Subject "subdivision of graph"

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Access status: Open Access ,
On incidence coloring of graph fractional powers
(Wydawnictwa AGH, 2023) Mozafari-Nia, Mahsa; Iradmusa, Moharram N.
For any $n \in \mathbb{N}$, the $n$-subdivision of a graph $G$ is a simple graph $G^\frac{1}{n}$ which is constructed by replacing each edge of $G$ with a path of length $n$. The $m$-th power of $G$ is a graph, denoted by $G^m$, with the same vertices of $G$, where two vertices of $G^m$ are adjacent if and only if their distance in $G$ is at most $m$. In [M.N. Iradmusa, <i>On colorings of graph fractional powers</i>, Discrete Math. 310 (2010), no. 10-11, 1551-1556] the $m$-th power of the $n$-subdivision of $G$, denoted by $G^\frac{m}{n}$ is introduced as a fractional power of $G$. The incidence chromatic number of $G$, denoted by $\chi_i(G)$, is the minimum integer $k$ such that $G$ has an incidence $k$-coloring. In this paper, we investigate the incidence chromatic number of some fractional powers of graphs and prove the correctness of the incidence coloring conjecture for some powers of graphs.