Abstract: The total chromatic number χ"(G) of a graph G is the
minimum number of colors needed to color the elements (vertices
and edges) of G such that no incident or adjacent pair of elements
receive the same color Let G be a graph with maximum degree Δ(G).
Considering a total coloring of G and focusing on a vertex with
maximum degree. A vertex with maximum degree needs a color and
all Δ(G) edges incident to this vertex need more Δ(G) + 1 distinct
colors. To color all vertices and all edges of G, it requires at least
Δ(G) + 1 colors. That is, χ"(G) is at least Δ(G) + 1. However,
no one can find a graph G with the total chromatic number which
is greater than Δ(G) + 2. The Total Coloring Conjecture states that
for every graph G, χ"(G) is at most Δ(G) + 2. In this paper, we prove that the Total Coloring Conjectur for a
Δ-claw-free 3-degenerated graph. That is, we prove that the total
chromatic number of every Δ-claw-free 3-degenerated graph is at
most Δ(G) + 2.
Abstract: In any production process, every product is aimed to attain a certain standard, but the presence of assignable cause of variability affects our process, thereby leading to low quality of product. The ability to identify and remove this type of variability reduces its overall effect, thereby improving the quality of the product. In case of a univariate control chart signal, it is easy to detect the problem and give a solution since it is related to a single quality characteristic. However, the problems involved in the use of multivariate control chart are the violation of multivariate normal assumption and the difficulty in identifying the quality characteristic(s) that resulted in the out of control signals. The purpose of this paper is to examine the use of non-parametric control chart (the bootstrap approach) for obtaining control limit to overcome the problem of multivariate distributional assumption and the p-value method for detecting out of control signals. Results from a performance study show that the proposed bootstrap method enables the setting of control limit that can enhance the detection of out of control signals when compared, while the p-value method also enhanced in identifying out of control variables.