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 $r$ be a ring with unity. the undirected nilpotent graph of $r$, denoted by $gamma_n(r)$, is a graph with vertex set ~$z_n(r)^* = {0neq x in r | xy in n(r) for some y in r^*}$, and two distinct vertices $x$ and $y$ are adjacent if and only if $xy in n(r)$, or equivalently, $yx in n(r)$, where $n(r)$ denoted the nilpotent elements of $r$. recently, it has been proved that if $r$ is a left ar...

A graph is planar if it can be drawn in the plane without edges crossing. More formally, a graph is planar if it has an embedding in the plane, in which each vertex is mapped to a distinct point P (v), and edge (u, v) to simple curves connecting P (u), P (v), such that curves intersect only at their endpoints. Examples of planar graphs: Pn, Trees, Cycles, X-tree, K4. Examples of non-planar grap...

In a book embedding of a graph G, the vertices of G are placed in order along a straight-line called spine of the book, and the edges of G are drawn on a set of half-planes, called the pages of the book, such that two edges drawn on a page do not cross each other. The minimum number of pages in which a graph can be embedded is called the book-thickness or the page-number of the graph. It is kno...

A graph G is called a 1-planar graph if it can be drawn on the plane so that each edge includes at most one crossing. It is not so difficult to see that |E(G)| ≤ 4|V (G)| − 8 for any 1-planar graph. In particular, a 1-planar graph G is said to be optimal if the equality holds. Suzuki has already proved that there exists an optimal 1-planar graph which can be embedded on the orientable closed su...

In 1988 Fellows conjectured that if a finite, connected graph admits a finite planar emulator, then it admits a finite planar cover. We construct a finite planar emulator for K4,5 − 4K2. Archdeacon [Dan Archdeacon, Two graphswithout planar covers, J. Graph Theory, 41 (4) (2002) 318–326] showed that K4,5−4K2 does not admit a finite planar cover; thusK4,5−4K2 provides a counterexample to Fellows’...

A graph is planar if it can be drawn in the plane without edges crossing. More formally, a graph is planar if it has an embedding in the plane, in which each vertex is mapped to a distinct point P (v), and edge (u; v) to simple curves connecting P (u); P (v), such that curves intersect only at their endpoints. Examples of planar graphs: Pn, Trees, Cycles, X-tree, K4. Examples of non-planar grap...

A graph is k-choosable if it can be colored whenever every vertex has a list of at least k available colors. A theorem by Grötzsch [2] asserts that every triangle-free planar graph is 3-colorable. On the other hand Voigt [10] found such a graph which is not 3-choosable. We prove that if a triangle-free planar graph is not 3-choosable, then it contains a 4-cycle that intersects another 4or 5-cyc...

A touching triangle graph (TTG) representation of a planar graph is a planar drawing Γ of the graph, where each vertex is represented as a triangle and each edge e is represented as a side contact of the triangles that correspond to the end vertices of e. We call Γ a proper TTG representation if Γ determines a tiling of a triangle, where each tile corresponds to a distinct vertex of the input g...

A graph G is k-choosable if for every assignment of a set S(v) of k colors to every vertex v of G, there is a proper coloring of G that assigns to each vertex v a color from S(v). We consider the complexity of deciding whether a given graph is k-choosable for some constant k. In particular, it is shown that deciding whether a given planar graph is 4-choosable is NP-hard, and so is the problem o...