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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
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.
According to Jensen & Toft (1995), the problem was first formulated by Nelson in 1950, and first published by Gardner (1960). Hadwiger (1945) had earlier published a related result, showing that any cover of the plane by five congruent closed sets contains a unit distance in one of the sets, and he also mentioned the problem in a later paper (Hadwiger 1961).
The target sum T in the SSP instance is set to an integer with "1" in the least-significant bit of every zone, that is, (2 0 +2 1 +...+2 3n-1). If the 3DM instance has a perfect matching, then summing the corresponding integers in the SSP instance yields exactly T. Conversely, if the SSP instance has a subset with sum exactly T, then, since the ...
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The summation of an explicit sequence is denoted as a succession of additions. For example, summation of [1, 2, 4, 2] is denoted 1 + 2 + 4 + 2, and results in 9, that is, 1 + 2 + 4 + 2 = 9. Because addition is associative and commutative, there is no need for parentheses, and the result is the same irrespective of the order of the summands ...