By Carol I.
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Additional info for A unified theory of elastic degradation and damage based on a loading surface
247-355. 5. J. C. Nash-Williams, Unexplored and semi-explored territories in graph theory, New Directions in th~ Theory of Graphs, Academic Press, New York, 1973, pp. 149-186. IRREDUCIBLE CONFIGURATIONS AND THE FOUR COLOR CONJECTURE Frank R. Bernhart Shippensburg State College Shippensburg, PA 17257 Abstract The classical four color reduction process takes on a new appearance in the light of the recently begun theory of open sets of colorings. In this paper we show that specific configurations and clusters can be simply classified as either reducible or irreducible, without appealing to the truth or falsity of the Four Color Conjecture By treating irreducibility and reducibility together, (4CC).
L • I and The vertex set of x If o the number of light edges directed out from x is at least and we write 1 • (x E D) Let x then t:he number of light edges incident to E L; directed toward x is at most h - 1. x and and we write (x E L) Since the directed dark edges define a bijection from we have (h-l) D to L. Hence ILl (h - 1) Inl :5: \' I xeD (number of directed light edges) - 2 \' L~ xEL The contradiction follows. Note that Theorem 1 shows immediately that there exists a multicoloring for the Petersen graph (Figure 1).
The process of reduction replaces a given cluster with something smaller, and usually highly degenerate. Figure 4 shows an example for a map cluster dotted line is where the circuit used to be) and the graph equivalent. (the Here we use an ordinary diagonal to mean a contact of two nonconsecutive ring elements, and we use a doubled diagonal (essentially an elongated equal sign) to indicate a merger of two ring elements. Single and double diagonals may also be treated as negative and positive edges of signed graphs, where two vertices joined by a negative (positive) edge have to have different (the same) colors in a coloring.
A unified theory of elastic degradation and damage based on a loading surface by Carol I.