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absorbing set adjacency matrix adjacent ann(x assume bipartite graphs cardinality chain graph characterisation chromatic number closed geodetic color column Combinatorial complete graph completes the proof components connected graph consider contains contradiction Corollary cycle define degree denote digraph dominating set domination number edge complete p,2 edges of G elements Ext(G Fibonacci Figure finite follows by Lemma graph G graph of order Graph Theory Hence implies independent set induced subgraph isomorphic L-coloring l)-labeling label Lemma Let G lottery numbers lottery set lower bound matrix maximum meet-semilattice n-set number of vertices obtain outerplanar graph p/d-graph path permutation graphs planar graph positive cut prime graph prove Ramsey numbers regular graph result follows semi-ideal sequence spectral radius square Subcase Suppose Szeged Theorem Theorem 3.1 tournament tree triple systems Tutte set ucgn(G unique upper bound vertex set vertices of G