147,477 research outputs found
Bilangan Terhubung Titik Pelangi Kuat Graf Octa-Chain (OCm)
An Octa-Chain graph (OCm) is a graph formed by modifying the cycle graph C8 by adding an edge connecting the midpoints in C8. The minimum number of colors used to color the vertices in a graph so that every two vertices have a rainbow path is called the rainbow vertex-connected number denoted by rvc (G). While the minimum number of colors used to color the vertices in a graph so that every two vertices are always connected by a rainbow path is called a strong rainbow vertex connected number and is denoted by srvc (G). This study aims to determine the rainbow vertex-connected number (rvc) and the strong rainbow-vertex-connected number (srvc) in the Octa-Chain graph (OCm). The results obtained from this research are the rainbow vertex-connected number rvc (OCm)=2m and the strong rainbow-vertex-connected number srvc (OCm)=2m
A Victorian Age Proof of the Four Color Theorem
In this paper we have investigated some old issues concerning four color map
problem. We have given a general method for constructing counter-examples to
Kempe's proof of the four color theorem and then show that all counterexamples
can be rule out by re-constructing special 2-colored two paths decomposition in
the form of a double-spiral chain of the maximal planar graph. In the second
part of the paper we have given an algorithmic proof of the four color theorem
which is based only on the coloring faces (regions) of a cubic planar maps. Our
algorithmic proof has been given in three steps. The first two steps are the
maximal mono-chromatic and then maximal dichromatic coloring of the faces in
such a way that the resulting uncolored (white) regions of the incomplete
two-colored map induce no odd-cycles so that in the (final) third step four
coloring of the map has been obtained almost trivially.Comment: 27 pages, 18 figures, revised versio
Using Markov chains to determine expected propagation time for probabilistic zero forcing
Zero forcing is a coloring game played on a graph where each vertex is
initially colored blue or white and the goal is to color all the vertices blue
by repeated use of a (deterministic) color change rule starting with as few
blue vertices as possible. Probabilistic zero forcing yields a discrete
dynamical system governed by a Markov chain. Since in a connected graph any one
vertex can eventually color the entire graph blue using probabilistic zero
forcing, the expected time to do this studied. Given a Markov transition matrix
for a probabilistic zero forcing process, we establish an exact formula for
expected propagation time. We apply Markov chains to determine bounds on
expected propagation time for various families of graphs
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