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3 problems match “graph theory”. Clear

Blue segments in a two-colored grid ●●●○○ 50 ptsA $16\times16$ square grid of dots is colored red and blue. Two dots that are adjacent in the grid (horizontally or vertically) are joined by a line segment. A segment joining two dots of the same color has that color, and a segment joining dots of different colors is yellow. There are 133 red dots in total. Exactly 2 of the 4 corner dots are red, and 32 other red dots lie on the boundary of the grid (not corners). There are 196 yellow segments in total. How many blue segments are there?
Matches played in a football tournament ●●●●○ 100 ptsA football invitational tournament involves sixteen cities, each sending two teams, Team A and Team B. Each pair of teams plays at most one match, and teams from the same city do not play each other. After several days, every team except Team A from City A has played a different number of matches. How many matches has Team B from City A played?
Maximum rounds guaranteeing a four-team condition ●●●●● 200 ptsThere are 18 teams in a single round-robin tournament. In each round the 18 teams are divided into 9 pairs, and the two teams in each pair play one match. The teams are regrouped from round to round, and over 17 rounds each team plays exactly one match against each of the other 17 teams. Suppose that after $n$ rounds, played according to any feasible schedule, there always exist 4 teams that have played exactly 1 match among themselves in total. Find the maximum possible value of $n$ .