The exact distance p-power of a graph G, denoted G [# p], is a graph on vertex set V (G) in
which two vertices are adjacent if they are at distance exactly p in G. Given integers k and p …

The exact distance t-power of a graph G, G [♯ t], is a graph which has the same vertex set as
G, with two vertices adjacent in G [♯ t] if and only if they are at distance exactly t in the …

For a graph G=(V, E), its exact-distance square, G [♯ 2], is the graph with vertex set V and
with an edge between vertices x and y if and only if x and y have distance (exactly) 2 in G …

An open packing in a graph G is a set S of vertices in G such that no two vertices in S have a
common neighbor in G. The injective chromatic number χ i (G) of G is the smallest number of …

In the past various distance based colorings on planar graphs were introduced. We turn our
focus to three of them, namely 2-distance coloring, injective coloring, and exact square …

A vertex coloring of a graph G=(V, E) is called an exact square coloring of G if any two
vertices at distance exactly 2 receive different colors. The minimum number of colors …

In this thesis, we study the 2-distance chromatic number of sparse graphs, namely, planar
graphs and graphs with bounded maximum average degree. Upper bounds are obtained by …

The $ k $-th exact-distance graph, of a graph $ G $ has $ V (G) $ as its vertex set, and $ xy $
as an edge if and only if the distance between $ x $ and $ y $ is (exactly) $ k $ in $ G $. We …

A vertex coloring of a graph G=(V, E) is called an exact square coloring of G if any pair of
vertices at distance 2 receive distinct colors. The minimum number of colors required by any …

It was recently pointed out that certain SiO $ _2 $ layer structures and SiO $ _2 $ nanotubes
can be described as full subdivisions aka subdivision graphs of partial cubes. A key tool for …