By Michael Stiebitz; et al
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Additional info for Graph edge coloring : Vizing's theorem and Goldberg's conjecture
He formulates it using a ( 2 n + l ) x ( 3 n + l)0-l -matrix with 1 -entries showing the colors possible for the edges from x. He then rearranges columns and rows of the matrix (changes the order of colors and of edges) and makes Kempe changes, corresponding to the arguments above, to see that the matrix may be changed into one with a 1 in all places (i, i) of the matrix, thus showing that it is possible to extend a coloring of G — x to include all edges from x also. Following Tait, König, and Shannon, the next breakthrough was the theorem of Vizing [297,298], obtained independently by Gupta .
Et} such that
then Fl will be extended to the multi-fan Ft+1 = (ei, j / i , . . , e»,i/i,ei+i, j/»+i), where e i + i = e' and j/ i + i is the endvertex of e' distinct from x. This can be done, since we allow a multi-fan to have multiple edges. Then we repeat the subroutine with i replaced by i + 1. If there is no such edge e', then Fl is a maximal multi-fan at x with respect to e and <£, and the algorithm returns the set Z = V(F1). Clearly, the algorithm stops after a finite number of steps, say in step p with F = FP = (ej ] 2/i, - ■ ·, 6p,2/p).
From this Shannon's theorem follows easily. Shannon's own proof for the case A(G) = 2r + 1 is by induction over the number of vertices. He removes a vertex x of degree 2 r + l fromG, colorsG — x by induction using 3r + 1 colors. Then he colors the edges incident at x one by one, using Kempe changes as the main tool. It would seem more appropriate to use induction on the number of edges. Let e € EG(X, y) be an edge of G and assume that G — e is edge-colored with 3r -I-1 colors by induction. We want to extend this coloring φ eC 3 r + 1 (G — e) by including the remaining edge e.
Graph edge coloring : Vizing's theorem and Goldberg's conjecture by Michael Stiebitz; et al