In this paper, we extend the Grötzsch Theorem by proving that the clique hypergraph H(G) of every planar graph is 3-colorable. We also extend this result to list colorings by proving that H(G) is 4-choosable for every planar or projective planar graph G. Finally, 4-choosability ofH(G) is established for the class of locally planar graphs on arbitrary surfaces.

Let G be a connected graph with maximum degree ∆. Brooks’ theorem states that G has a ∆-coloring unless G is a complete graph or an odd cycle. A graph G is degree-choosable if G can be properly colored from its lists whenever each vertex v gets a list of d(v) colors. In the context of list coloring, Brooks’ theorem can be strengthened to the following. Every connected graph G is degree-choosabl...

A majority coloring of a digraph is a coloring of its vertices such that for each vertex v, at most half of the out-neighbors of v have the same color as v. A digraph D is majority k-choosable if for any assignment of lists of colors of size k to the vertices there is a majority coloring of D from these lists. We prove that every digraph is majority 4-choosable. This gives a positive answer to ...

A proper total weighting of a graph G is mapping φ which assigns to each vertex and edge real number as its weight so that for any uv , Σ e ∈ E ( v ) φ( )+φ( ≠ u ). k,k ')-list assignment L set k permissible weights ' weights. An -total with z V ∪ called ')-choosable if every there exists weighting. As strenghtening the well-known 1-2-3 conjecture, it was conjectured in [Wong Zhu, Total choosab...

We introduce a notion of color-criticality in the context chromatic-choosability. define graph $G$ to be strong $k$-chromatic-choosable if $\chi(G) = k$ and every $(k-1)$-assignment for which is not list-colorable has property that lists are same all vertices. That usual coloring is, some sense, obstacle list-coloring. prove basic properties strongly chromatic-choosable graphs such as chromatic...

An incidence of a graph G is a pair (v, e) where v is a vertex of G and e is an edge of G incident with v. Two incidences (v, e) and (w, f) of G are adjacent whenever (i) v = w, or (ii) e = f , or (iii) vw = e or f . An incidence p-colouring of G is a mapping from the set of incidences of G to the set of colours {1, . . . , p} such that every two adjacent incidences receive distinct colours. In...

This paper discusses the circular version of list coloring of graphs. We give two definitions of the circular list chromatic number (or circular choosability) of a graph and prove that they are equivalent. Then we prove that for any graph , . Examples are given to show that this bound is sharp in the sense that for any , there is a graph with . It is also proved that -degenerate graphs have . T...

A vertex colouring of a graph G is nonrepetitive if for any path P = (v1, v2, . . . , v2r) in G, the first half is coloured differently from the second half. The Thue choice number of G is the least integer l such that for every l-list assignment L of G, there exists a nonrepetitive L-colouring of G. We prove that for any positive integer l, there is a tree T with πch(T ) > l. On the other hand...