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4-Coloring 4-connected adjacent alternating group graph APX-complete bottleneck flow control color common vertex complete the path CONGRESSUS NUMERANTIUM 175 connected dominating set connected graph consider Corollary coset table cosets defined denote digraph disjoint paths domination graph edges elements exponential generating function Figure floorplanning gameboard Ramsey numbers Genocchi Genocchi numbers given graph G graph of order Graph Theory grid graphs Hamiltonian number Hamiltonian walk Hankel matrix Hedetniemi Hence interval graph Lemma Let G Let us assume matrix maximal cliques minimum broadcasting minimum connected dominating minSGSP monochromatic P3 Nash equilibrium point neighboring substars netlist number of nodes number of vertices obtained optimization P-complete pair path from S2 permutations problem processor Proof Proposition prove route RRDFs s2 to t2 sequence SGC graph SGCP star graph steps sub-blocks subgraph of G substars H2 Suppose tournament update upper bound variants vertex virtual circuits Whitney numbers