46 research outputs found
On the Set of Circular Total Chromatic Numbers of Graphs
For every integer and every \eps>0 we construct a graph with
maximum degree whose circular total chromatic number is in the interval
(r,r+\eps). This proves that (i) every integer is an accumulation
point of the set of circular total chromatic numbers of graphs, and (ii) for
every , the set of circular total chromatic numbers of graphs with
maximum degree is infinite. All these results hold for the set of
circular total chromatic numbers of bipartite graphs as well
Locating and Identifying Codes in Circulant Networks
A set S of vertices of a graph G is a dominating set of G if every vertex u
of G is either in S or it has a neighbour in S. In other words, S is dominating
if the sets S\cap N[u] where u \in V(G) and N[u] denotes the closed
neighbourhood of u in G, are all nonempty. A set S \subseteq V(G) is called a
locating code in G, if the sets S \cap N[u] where u \in V(G) \setminus S are
all nonempty and distinct. A set S \subseteq V(G) is called an identifying code
in G, if the sets S\cap N[u] where u\in V(G) are all nonempty and distinct. We
study locating and identifying codes in the circulant networks C_n(1,3). For an
integer n>6, the graph C_n(1,3) has vertex set Z_n and edges xy where x,y \in
Z_n and |x-y| \in {1,3}. We prove that a smallest locating code in C_n(1,3) has
size \lceil n/3 \rceil + c, where c \in {0,1}, and a smallest identifying code
in C_n(1,3) has size \lceil 4n/11 \rceil + c', where c' \in {0,1}
On the uniquely list colorable graphs
Let G be a graph with n vertices and suppose that for each vertex v in G, there exists a list of k colors, L(v), such that there is a unique proper coloring for G from this collection of lists, then G is called a uniquely k–list colorable graph. Recently M. Mahdian and E.S. Mahmoodian characterized uniquely 2–list colorable graphs. Here we state some results which will pave the way in characterization of uniquely k–list colorable graphs. There is a relationship between this concept and defining sets in graph colorings and critical sets in latin squares. 1 Introduction an