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A fall coloring of a graph G is a proper coloring of the vertex set of G such that every vertex of G is a color dominating vertex in G (that is, it has at least one neighbor in each of the other color classes). The fall coloring number ${\chi }_f\left(G\right)$ of G is the minimum size of a fall color partition of G (when it exists). Trivially, for any graph G, $\chi \left(G\right)\le {\chi }_f\left(G\right)$. In this paper, we show the existence of an infinite family of graphs G with prescribed values for χ(G) and ${\chi }_f\left(G\right)$. We also obtain the smallest non-fall colorable...

The Wiener number of a graph G is defined as $1/2{\sum }_{u,v\in V\left(G\right)}d(u,v)$, d the distance function on G. The Wiener number has important applications in chemistry. We determine a formula for the Wiener number of an important graph family, namely, the Mycielskians μ(G) of graphs G. Using this, we show that for k ≥ 1, $W\left(\mu \left(S{ₙ}^k\right)\right)\le W\left(\mu \left(T{ₙ}^k\right)\right)\le W\left(\mu \left(P{ₙ}^k\right)\right)$, where Sₙ, Tₙ and Pₙ denote a star, a general tree and a path on n vertices respectively. We also obtain Nordhaus-Gaddum type inequality for the Wiener number of $\mu \left(G^k\right)$.

The Wiener number of a graph G is defined as 1/2∑d(u,v), where u,v ∈ V(G), and d is the distance function on G. The Wiener number has important applications in chemistry. We determine the Wiener number of an important family of graphs, namely, the Kneser graphs.