1,448 research outputs found
On a Vizing-type integer domination conjecture
Given a simple graph , a dominating set in is a set of vertices
such that every vertex not in has a neighbor in . Denote the domination
number, which is the size of any minimum dominating set of , by .
For any integer , a function
is called a \emph{-dominating function} if the sum of its function
values over any closed neighborhood is at least . The weight of a
-dominating function is the sum of its values over all the vertices. The
-domination number of , , is defined to be the
minimum weight taken over all -domination functions. Bre\v{s}ar,
Henning, and Klav\v{z}ar (On integer domination in graphs and Vizing-like
problems. \emph{Taiwanese J. Math.} {10(5)} (2006) pp. 1317--1328) asked
whether there exists an integer so that . In this note we use the Roman -domination number,
of Chellali, Haynes, Hedetniemi, and McRae, (Roman
-domination. \emph{Discrete Applied Mathematics} {204} (2016) pp.
22-28.) to prove that if is a claw-free graph and is an arbitrary
graph, then , which also implies the conjecture for all .Comment: 8 page
- …