Connected counterexamples to the exact color-critical two-eigenvalue bound
For F=K5 minus one edge, a color-critical graph of chromatic number four, the graph formed from K(s,s,s) and K4 by one bridge is connected and F-free for every integer s at least three. It has 3s squared plus 7 edges, and the sum of squares of its two largest ordered adjacency eigenvalues exceeds 4m/3 by at least (656s squared-1366s+62)/(3(37s-1) squared)>0. The family has arbitrarily large edge counts, refuting the exact assertion in KKP Conjecture 1.16 and Li–Liu–Zhang Conjecture 5.4.
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