A Note on Maxflow-Mincut and Homomorphic Equivalence in Matroids
نویسنده
چکیده
Graph homomorphisms are used to study good characterizations for coloring problems (Trans. Amer. Math. Soc. 384 (1996), 1281–1297; Discrete Math. 22 (1978), 287–300). Particularly, the following concept arises in this context: A pair of graphs (A, B) is called a homomorphism duality if for any graph G either there exists a homomorphism σ : A→ G or there exists a homomorphism τ : G → B but not both. In this paper we show that maxflow-mincut duality for matroids can be put into this framework using strong maps as homomorphisms. More precisely, we show that, if Ck denotes the circuit of length k+1, the pairs (Ck ,Ck+1) are the only homomorphism dualities in the class of duals of matroids with the strong integer maxflow-mincut property (Jour. Comb. Theor. Ser.B 23 (1977), 189–222). Furthermore, we prove that for general matroids there is only a trivial homomorphism duality.
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