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The sum of the labels is 11, smaller than could be achieved using only two labels. In graph theory, a sum coloring of a graph is a labeling of its vertices by positive integers, with no two adjacent vertices having equal labels, that minimizes the sum of the labels. The minimum sum that can be achieved is called the chromatic sum of the graph. [1]
The criterion of minimalization is the sum of colors Star coloring Every 2-chromatic subgraph is a disjoint collection of stars Strong coloring Every color appears in every partition of equal size exactly once Strong edge coloring Edges are colored such that each color class induces a matching (equivalent to coloring the square of the line graph)
In mathematics, an empty sum, or nullary sum, [1] is a summation where the number of terms is zero. The natural way to extend non-empty sums [ 2 ] is to let the empty sum be the additive identity . Let a 1 {\displaystyle a_{1}} , a 2 {\displaystyle a_{2}} , a 3 {\displaystyle a_{3}} , ... be a sequence of numbers, and let
Minkowski sums act linearly on the perimeter of two-dimensional convex bodies: the perimeter of the sum equals the sum of perimeters. Additionally, if K {\textstyle K} is (the interior of) a curve of constant width , then the Minkowski sum of K {\textstyle K} and of its 180° rotation is a disk.
A root (or zero) of a chromatic polynomial, called a “chromatic root”, is a value x where (,) =. Chromatic roots have been very well studied, in fact, Birkhoff’s original motivation for defining the chromatic polynomial was to show that for planar graphs, P ( G , x ) > 0 {\displaystyle P(G,x)>0} for x ≥ 4.
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That is, G 1 and G 2 can be represented on the same set of n vertices with no edges in common. The Hajnal–Szemerédi theorem is the special case of this conjecture in which G 2 is a disjoint union of cliques. Catlin (1974) provides a similar but stronger condition on Δ 1 and Δ 2 under which such a packing can be guaranteed to exist.
The reciprocals of the non-negative integer powers of 2 sum to 2 . This is a particular case of the sum of the reciprocals of any geometric series where the first term and the common ratio are positive integers. If the first term is a and the common ratio is r then the sum is r / a (r − 1) .