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] gave such a graph which is not 3-choosable. We prove that every triangle-free planar graph such that 4-cycles do not share edges with other 4and 5-cycles is 3-choosable. T...

Given a planar graph G, what is the maximum number of collinear vertices in a planar straight-line drawing of G? This problem resides at the core of several graph drawing problems, including universal point subsets, untangling, and column planarity. The following results are known: Every n-vertex planar graph has a planar straight-line drawing with Ω( √ n) collinear vertices; for every n, there...

let $a$ be a non-trivial abelian group and $a^{*}=asetminus {0}$. a graph $g$ is said to be $a$-magic graph if there exists a labeling$l:e(g)rightarrow a^{*}$ such that the induced vertex labeling$l^{+}:v(g)rightarrow a$, define by $$l^+(v)=sum_{uvin e(g)} l(uv)$$ is a constant map.the set of all constant integerssuch that $sum_{uin n(v)} l(uv)=c$, for each $vin n(v)$,where $n(v)$ denotes the s...

A graph is called 1-planar if it can be drawn in the plane so that each of its edges is crossed by at most one other edge. We show that every 1-planar drawing of any 1-planar graph on n vertices has at most n − 2 crossings; moreover, this bound is tight. By this novel necessary condition for 1-planarity, we characterize the 1-planarity of Cartesian product Km × Pn. Based on this condition, we a...

A topological drawing of a graph is fan-planar if for each edge e the edges crossing e have a common endpoint on the same side of e, and a fan-planar graph is a graph admitting such a drawing. Equivalently, this can be formulated by two forbidden patterns, one of which is the configuration where e is crossed by two independent edges and the other where e is crossed by incident edges with the co...

The notion of a planar st graph (also known as e-bipolar planar graph) is essentially equivalent to that of a progressive plane graph, which was introduced by Joyal and Street in the theory of graphical calculus for tensor categories. Fraysseix and Mendez have shown a bijection between equivalence classes of planar st embeddings of a directed graph G and the conjugate orders of the edge poset o...

We prove that we can specify by formulas of monadic second-order logic the unique planar embedding of a 3-connected planar graph. If the planar graph is not 3-connected but given with a linear order of its set of edges, we can also define a planar embedding by monadic second-order formulas. We cannot do so in general without the ordering, even for 2-connected planar graphs. The planar embedding...

Any simple planar graph can be triangulated, i.e., we can add edges to it, without adding multi-edges, such that the result is planar and all faces are triangles. In this paper, we study the problem of triangulating a planar graph without increasing the pathwidth by much. We show that if a planar graph has pathwidth k, then we can triangulate it so that the resulting graph has pathwidth O(k) (w...

A graph G is called “planar” if there is a way to draw it in the plane (e.g. on a piece of paper) such that there are no crossings among edges, except of course of the endpoints of edges which may coinside upon common vertices (shared by more than one edge). Given in another way: a graph G is planar if a way exists to draw it in the plane such that any of the plane's points are at most occupied...

Planar graphs originated with the studies of polytopes and of maps. The skeleton (edges) of a threedimensional polytope provide a planar graph. We obtain a planar graph from a map by representing countries by vertices, and placing edges between countries that touch each other. Assuming each country is contiguous, this gives a planar graph. While planar graphs were introduced for practical reaso...