n = 37  edges = 263  sum deg = 526
degree multiset (desc): [29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 13, 13, 13, 13, 13, 13, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]
w (distinct degrees) = 30   n - w = 7
not 2-colourable
3-colourable; proper colouring verified on all 263 edges; class sizes [7, 12, 18]
chi = 3;  ceil(floor(w/2)/(n-w)) = ceil(15/7) = 3;  conjecture requires chi > 3: False
equivalent form (2chi-1)(n-w) >= n-1 : 35 >= 36 -> False
