### Table 3: The worst con gurations for 7-colouring 4-regular triangle-free graphs.

1998

"... In PAGE 9: ... Let i be the number of nonisomorphic rim colourings of the ith con guration, 1 i 15. These values are listed in the fth column of Table 1, while a complete list of the nonisomorphic rim colourings for each con guration can be found in Table3 of the Appendix. Each rim colouring is shown as a list of six numbers representing the colours of the six rim vertices in the order given above.... In PAGE 12: ... It was necessary to consider 42574 non-isomorphic rim colourings. The worst cases for the di erent values of (w) are listed in Table3 . Each rim colouring is shown as a list of 12 numbers in four groups of three.... In PAGE 13: ...bound over (w) in Table3 .... ..."

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### Table 10. Number of Backtrack Nodes to Solve Various instances in triangle-free graphs

1999

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### Table 5.1 The worst configurations for 7-coloring 4-regular triangle-free graphs.

1999

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### Table 2. Numbers of triangle-free graphs The size of a graph is the number of edges in it. One way to view the one-color triangle avoidance game is that A apos;s objective is to reach a maximal triangle-free graph of odd size, whereas B apos;s goal is to reach one of even size. We call these sets of graphs the natural objectives of A and B. As noted earlier in describing our computational procedure, maximal triangle-free graphs are exactly those which still have a mark value of U when reached in the main loop. In this way we have counted maximal triangle-free graphs in the course of determining the Winners for n 12. The number mn of order n is listed for those values of n in Table 2, along with the number wn which are Winner apos;s objective. We attempted to gain insight into Winner apos;s strategy by modifying the game so as to limit Winner apos;s objectives to some proper subset of her natural 9

"... In PAGE 9: ... Both time and space posed barriers to extending the computations to order n = 13. The numbers tn of unlabeled triangle-free graphs of order n as reported in [5] are listed in Table2 for n 15. The time and the space requirements for our computational procedure for determining Winner for... ..."

### Table 1 gives the numbers of MT F s with up to 20 vertices. Up to 17 vertices the lists were checked by a completely independent generation program for MT F s and up to 14 vertices they were additionally checked by generating all triangle free graphs using the graph generator makeg [14] and ltering the output for graphs with diameter 2.

"... In PAGE 2: ... Table1 : Numbers of MT F s Since the number of MT F s is not too large for small vertex numbers, for small triangle Ramsey numbers one could simply apply Lemma 1. Obviously no connected graph of order n can be... ..."

### Table 3. Number of edges in Canonical Critical Subgraphs of the triangle-free graphs. The canonical critical graph is obtained by deleting as many edges as possible choosing them in vertex order. The size of critical sets and subgraphs are shown in tables 3 and 4. We can do this only up to n = 125 because of the computational expense. As n increases in Table 3, we see that the size of critical subgraphs converges at the rst frozen point and threshold. The size of these graphs grows at least linearly with n, suggesting that critical graphs have size O(n). The size of critical sets in Table 4

1999

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### Table 1: DIMACS graph coloring bench-

2004

"... In PAGE 21: ... We use two families in this work, named mulsol, zeroin Mycielski graphs. Instances of triangle-free graphs based on the Mycielski (My- cielski, 1955) transformation, called myciel Table1 gives the name, size (number of vertices and edges) and the chromatic number for each benchmark. We use a maximum value of K = 20 for K coloring.... In PAGE 23: ...0e+164 941 167 NU+SC 437K 777925 3193 5.0e+148 597 47 Table 2: CNF formula sizes, symmetry detection results and runtimes, totaled for 20 benchmarks from Table1 , with K = 20. NU = null-color elimina- tion; CA = cardinality-based; LI = lowest-index; SC = selective coloring.... ..."

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### Table 1: DIMACS graph coloring bench-

2004

"... In PAGE 21: ... We use two families in this work, named mulsol, zeroin Mycielski graphs. Instances of triangle-free graphs based on the Mycielski (My- cielski, 1955) transformation, called myciel Table1 gives the name, size (number of vertices and edges) and the chromatic number for each benchmark. We use a maximum value of K = 20 for K coloring.... In PAGE 23: ...0e+164 941 167 NU+SC 437K 777925 3193 5.0e+148 597 47 Table 2: CNF formula sizes, symmetry detection results and runtimes, totaled for 20 benchmarks from Table1 , with K = 20. NU = null-color elimina- tion; CA = cardinality-based; LI = lowest-index; SC = selective coloring.... ..."

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### TABLE 2. Upper Bounds on Size of Induced Trees and Forests, a.a.s.

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### Table 3: Average proportion of vertices in induced planar subgraph found in randomly generated graphs of 10,000 vertices. (Standard deviations in parenthesis.)

"... In PAGE 18: ... The behaviour of the other algorithms as d increases in terms of the reduction in proportion of vertices in the induced subgraph is re- markably similar as can be seen in Figure 1. In Table3 the proportion of vertices in subgraphs of graphs of 10,000 vertices are displayed. As n increases, the proportion of vertices in the subgraphs of 6-regular graphs and graphs of expected average degree 6 found by the various MIPS algorithms quickly converge to these proportions.... In PAGE 18: ... Each of the algorithms converges, in this sense, on or before n = 200, except for the HL algorithm on graphs of expected average degree d. The proportions of vertices in the subgraphs found by the each of these algorithms difier by only a few percent in smaller graphs, but as n becomes large only by a fraction of a percent (see Table3 ). The fact that the average sizes of subgraph produced by these algorithms are so similar even though the classes of induced subgraphs produced are not all the same may suggest that there may be some fundamental limit on the performance of algorithms for flnding induced subgraphs with many hereditary properties.... In PAGE 18: ... The fact that the average sizes of subgraph produced by these algorithms are so similar even though the classes of induced subgraphs produced are not all the same may suggest that there may be some fundamental limit on the performance of algorithms for flnding induced subgraphs with many hereditary properties. The size of subgraph produced by the algorithms (excluding the HL and IS algorithms) was very similar: the range of sizes of sub- graph produced varied only by a few percent (see Table3 ). Usually,... ..."

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