{"id":"palomar/PALOMAR-2026-10-16-000012","url":"https://github.com/shaikidris/kkp-connected-spectral-counterexamples/tree/c611a4a733fe796809cd941f96ff8f2c294fd1ef","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-10-16-000012-v1.json","repo":"https://github.com/shaikidris/kkp-connected-spectral-counterexamples","tags":["math.CO","msc:05C50","msc:05C35","extremal-graph-theory","spectral-graph-theory","counterexample"],"score":0,"title":"Connected counterexamples to the exact color-critical two-eigenvalue bound","trust":"high","domain":"math.CO","source":"palomar","authors":["Idris Ali Shaik"],"reviews":0,"subject":"math.CO","summary":"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.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-10-16","reactions":{},"formalized":true,"formalizes":[{"title":"Asymptotic rank two spectral bounds and a color critical counterexample","identifier":"","relationship":"formalizes"}]}