### Table 1: Characteristic expressions for all possible types of color comparisons of vertices of weights at most 2 for the three-color plurality problem.

2006

"... In PAGE 12: ... However, if color(u) = color(v0) and color(u0) 6 = color(v), an edge is added between u and v and the weight of u increases from 1 to 2; similarly, if color(u0) = color(v) and color(u) 6 = color(v0), an edge is added between u and v and the weight of v increases from 1 to 2. In other words, 2w1 = 1 42w1 + 1 4w2 + 1 2(w1 +w2); which gives the characteristic expression w1 + 3 4w2: Table1 gives the characteristic expressions for all possible color comparisons when c = 3; that is, for the nine types of color comparisons involving vertices of weights 0, 1, and 2. Color comparisons of types 120 and 220 are of no interest because their characteristic expressions are 0, and hence they do not cause any change in the weights of the vertices.... In PAGE 12: ... Denote by Nij the average number of times that color comparisons of type ij occur during the algorithm, and denote by Nij0 the corresponding value for color comparisons of type ij0. Because we start with n vertices of weight 0 and each color comparison of type 00 consumes an average of 5/3 vertices of weight 0 (see the entries in the rightmost column of Table1 ), while each type 01 or 02 color comparison consumes a single vertex of weight 0, at the end of the algorithm the expected number of remaining vertices with weight 0 is W0 = n 5 3N00 N01 N02: (9) We similarly count the average number W1 of vertices of weight 1 at the end of the algorithm and we have W1 = 4 3N00 + 1 3N01 + 2 3N02 3 2N11 N110 N12 (10)... In PAGE 14: ... De ne a block to be two vertices of degree 1 connected to each other by edge that is, they are disconnected from the other vertices. All of the characteristic expressions in Table1 are correct, except type 00 which now causes a new block. The expected effect of a type 00 color comparison is now 2w0 = 1 3w0 + 2 3wb;... ..."

### Table 2 The performance of the coloring heuristics. Heuristic Problem Number of colors

in Parallel Iterative Solution Of Sparse Linear Systems Using Orderings From Graph Coloring Heuristics

1990

"... In PAGE 4: ... The rst experiment shows the performance of the coloring heuristics on LAP5 and LAP9 for which an optimal coloring is known. The results in Table2 show that both algorithms produce optimal or slightly suboptimal colorings but the IDO heuristic is slightly superior. In the second set of experiments, the performance of the coloring heuristics is compared with three other ordering algorithms: minimum degree, reverse Cuthill- McKee (RCM), and nested dissection.... ..."

Cited by 3

### Table 5: Color Codes of the Default Color Map 3.6 Further Remarks A few important restrictions should be considered. All titles of graphs and nodes must be unique. In order to decide which are the source and the target node of an edge, this restriction is very important. A node can only be touched by 2 near edges. If more than 2 near edges are speci ed to touch the node, the remaining near edges are converted into normal edges. A node that has anchored edges can have only maximal 1 near edge. Further, if anchored edges occur, the orientation is always top_to_bottom. It is possible to change the colors or underline during the output of text, e.g., drawing of labels or info elds. This is controlled by special characters in the corresponding string

1995

"... In PAGE 29: ... If we now specify color: blue, then the color khaki will appear. Table5 shows the default... ..."

Cited by 1

### Table 1 lists the complexity results on the problems considered here, for graphs with few cliques. It is interesting to note that the NP-completeness of edge clique cover implies that it remains hard to find a minimum intersection representation even when given a minimum Helly intersection representation. The problems that admit efficient algorithms on these restricted classes appear to be only those problems where the restriction allows a brute force algorithm to run efficiently. Whether there is a problem that allows this restriction to be used in the construction of a nontrivial efficient algorithm is unknown.

2005

### Table 1: Results on hard graph coloring problems

"... In PAGE 5: ... These problems have been shown to defeat many algorithmic complete search methods. Table1 shows the average timings required for each of our tested MGAs to solve these prob- lems. The MCH-MGA0 improved with popu learn and lazy look forward performs the best among the other MGAs since the lazy look-forward algorithm is invoked less frequently than the full look-forward algorithm.... ..."

### Table 1: Results on hard graph coloring problems

"... In PAGE 6: ... These problems have been shown to defeat many algorithmic complete search methods. Table1 shows the average timings required for each of our tested MGAs to solve these prob- lems. The MCH-MGA0 improved with popu learn and lazy look forward performs the best among the other MGAs since the lazy look-forward algorithm is invoked less frequently than the full look-forward algorithm.... ..."

### Table 1: Results on hard graph coloring problems

"... In PAGE 5: ... These problems have been shown to defeat many algorithmic complete search methods. Table1 shows the average timings required for each of our tested MGAs to solve these prob- lems. The MCH-MGA0 improved with popu learn and lazy look forward performs the best among the other MGAs since the lazy look-forward algorithm is invoked less frequently than the full look-forward algorithm.... ..."

### Table 4: Color Inverse Halftoning independently from Color Planes

"... In PAGE 4: ... Other types of halftoning methods can also be applied to these images. In Table4 we show the PSNR values of the reconstructed images. We have included Lena, peppers, Ti any, mandrill and sailboat in the training set.... ..."

### Table 2: Efficiency of KTS algorithm on problems of different condition numbers.

2007

### Table 1. A recurrence arising from unpublished work with J. Byskov on graph coloring algorithms.

2003

"... In PAGE 2: ... Similar but somewhat more complicated multivariate recurrences have arisen in our algorithm for 3-coloring [10] with variables counting 3- and 4-value variables in a constraint satisfaction instance, and in our algorithm for the traveling salesman problem in cubic graphs [12] with variables counting vertices, unforced edges, forced edges, and 4-cycles of unforced edges. These examples of recurrences have all had few enough terms that they can be solved by hand, but Table1 depicts a recurrence, arising from unpublished work with J. Byskov on graph coloring algorithms, that is complex enough that hand solution seems unlikely.... In PAGE 8: ... Both due to our use of exact real arithmetic, and due to the implementation in Python, a relatively slow interpreted language, our exact arithmetic implementation is not fast, taking several hours on a laptop computer to solve moderately sized 3-variable recurrences to 64 bits of precision. However our floating point implementation is able to run at interactive speeds, taking roughly one or two seconds on a recent- model laptop to solve recurrences such as the one in Table1... In PAGE 11: ... Perhaps it would also be possible to automatically perform some of the case analysis used to design backtracking algorithms, and to determine the appropriate variables to use in setting up the recur- rences used to analyze those algorithms, before automatically solving those recurrences, at least for simple constraint satisfaction type problems. It would also be of interest to find ways of specifying algorithms of this type in such a way that their correctness can be proven automatically, especially in situations where re- peated refinement based on our analysis tools has led to highly complex case analysis such as that appearing in Table1 . Also, while we can find tight worst-case bounds on the solution of the recurrence derived from an algorithm, it may not always be possible to construct an instance causing the algorithm itself to have that worst case time bound; it would be useful to determine conditions under which this recurrence-based analysis is tight.... ..."