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algorithm balls per cell BASIS STEP chromatic number coefficient of xr colors combinatorial connected graph Consider Figure contains counting degree determining the number dice digit directed graph distinct cells elements equivalent Eulerian circuit Example Exercises exponential generating function factor Find a recurrence given in Figure graph G graph in Figure Graph Theory Hamiltonian cycle Hamiltonian path hence identical balls inclusion-exclusion principle induction hypothesis induction step initial conditions integer equation integer solutions intersection isomorphic least mathematical induction minimal spanning tree multiple multiset N(Ai natural numbers number of balls number of edges number of integer number of solutions number of vertices number of zeros objects obtain odd number one-one correspondence pair planar simple graph polynomial postfix expressions problem proof Proposition Prove recurrence relation rule of product rule of sum Show by induction simple graph solutions to Xi subsets techniques true vertex weighted graph well-ordering property