31 research outputs found
Characterization of Randomly k-Dimensional Graphs
For an ordered set of vertices and a vertex in a
connected graph , the ordered -vector
is called the (metric) representation
of with respect to , where is the distance between the vertices
and . The set is called a resolving set for if distinct vertices
of have distinct representations with respect to . A minimum resolving
set for is a basis of and its cardinality is the metric dimension of
. The resolving number of a connected graph is the minimum , such
that every -set of vertices of is a resolving set. A connected graph
is called randomly -dimensional if each -set of vertices of is a
basis. In this paper, along with some properties of randomly -dimensional
graphs, we prove that a connected graph with at least two vertices is
randomly -dimensional if and only if is complete graph or an
odd cycle.Comment: 12 pages, 3 figure
Chromatic Equivalence Classes and Chromatic Defining Numbers of Certain Graphs
There are two parts in this dissertation: the chromatic equivalence classes and
the chromatic defining numbers of graphs.
In the first part the chromaticity of the family of generalized polygon trees with
intercourse number two, denoted by Cr (a, b; c, d), is studied. It is known that
Cr( a, b; c, d) is a chromatic equivalence class if min {a, b, c, d} ≥ r+3. We consider
Cr( a, b; c, d) when min{ a, b, c, d} ≤ r + 2. The necessary and sufficient conditions
for Cr(a, b; c, d) with min {a, b, c, d} ≤ r + 2 to be a chromatic equivalence class
are given. Thus, the chromaticity of Cr (a, b; c, d) is completely characterized.
In the second part the defining numbers of regular graphs are studied. Let
d(n, r, X = k) be the smallest value of defining numbers of all r-regular graphs
of order n and the chromatic number equals to k. It is proved that for a given
integer k and each r ≥ 2(k - 1) and n ≥ 2k, d(n, r, X = k) = k - 1. Next,
a new lower bound for the defining numbers of r-regular k-chromatic graphs
with k < r < 2( k - 1) is found. Finally, the value of d( n , r, X = k) when
k < r < 2(k - 1) for certain values of n and r is determined