Colouring planar mixed hypergraphs.
About uniquely colorable mixed hypertrees
A mixed hypergraph is a triple 𝓗 = (X,𝓒,𝓓) where X is the vertex set and each of 𝓒, 𝓓 is a family of subsets of X, the 𝓒-edges and 𝓓-edges, respectively. A k-coloring of 𝓗 is a mapping c: X → [k] such that each 𝓒-edge has two vertices with the same color and each 𝓓-edge has two vertices with distinct colors. 𝓗 = (X,𝓒,𝓓) is called a mixed hypertree if there exists a tree T = (X,𝓔) such that every 𝓓-edge and every 𝓒-edge induces a subtree of T. A mixed hypergraph 𝓗 is called uniquely...
Color-bounded hypergraphs, V: host graphs and subdivisions
A color-bounded hypergraph is a hypergraph (set system) with vertex set X and edge set = E₁,...,Eₘ, together with integers and satisfying for each i = 1,...,m. A vertex coloring φ is proper if for every i, the number of colors occurring in edge satisfies . The hypergraph ℋ is colorable if it admits at least one proper coloring. We consider hypergraphs ℋ over a “host graph”, that means a graph G on the same vertex set X as ℋ, such that each induces a connected subgraph in G. In the current...
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