Related papers: On uniquely 3-colorable plane graphs without presc…
A graph $G$ is \emph{uniquely k-colorable} if the chromatic number of $G$ is $k$ and $G$ has only one $k$-coloring up to permutation of the colors. A uniquely $k$-colorable graph $G$ is edge-critical if $G-e$ is not a uniquely $k$-colorable…
A plane graph is l-facially k-colourable if its vertices can be coloured with k colours such that any two distinct vertices on a facial segment of length at most l are coloured differently. We prove that every plane graph is 3-facially…
A facial-parity edge-coloring of a $2$-edge-connected plane graph is a facially-proper edge-coloring in which every face is incident with zero or an odd number of edges of each color. A facial-parity vertex-coloring of a $2$-connected plane…
A facial unique-maximum coloring of a plane graph is a vertex coloring where on each face $\alpha$ the maximal color appears exactly once on the vertices of $\alpha$. If the coloring is required to be proper, then the upper bound for the…
An $\ell$-facial edge-coloring of a plane graph is a coloring of its edges such that any two edges at distance at most $\ell$ on a boundary walk of any face receive distinct colors. It is the edge-coloring variant of the $\ell$-facial…
A facial unique-maximum coloring of a plane graph is a proper coloring of the vertices using positive integers such that each face has a unique vertex that receives the maximum color in that face. Fabrici and G\"{o}ring (2016) proposed a…
We study straight-line drawings of graphs where the vertices are placed in convex position in the plane, i.e., \emph{convex drawings}. We consider two families of graph classes with convex drawings: \emph{outer $k$-planar} graphs, where…
A graph G is (d_1,..,d_l)-colorable if the vertex set of G can be partitioned into subsets V_1,..,V_l such that the graph G[V_i] induced by the vertices of V_i has maximum degree at most d_i for all 1 <= i <= l. In this paper, we focus on…
We show that every plane graph with maximum face size four whose all faces of size four are vertex-disjoint is cyclically 5-colorable. This answers a question of Albertson whether graphs drawn in the plane with all crossings independent are…
A facial parity edge coloring of a 2-edge connected plane graph is an edge coloring where no two consecutive edges of a facial walk of any face receive the same color. Additionally, for every face f and every color c either no edge or an…
The Four Colour Theorem asserts that the vertices of every plane graph can be properly coloured with four colors. Fabrici and G\"oring conjectured the following stronger statement to also hold: the vertices of every plane graph can be…
A facial $3$-complete $k$-coloring of a triangulation $G$ on a surface is a vertex $k$-coloring such that every triple of $k$-colors appears on the boundary of some face of $G$. The facial $3$-achromatic number $\psi_3(G)$ of $G$ is the…
Considering regions in a map to be adjacent when they have nonempty intersection (as opposed to the traditional view requiring intersection in a linear segment) leads to the concept of a facially complete graph: a plane graph that becomes…
The only open case of Vizing's conjecture that every planar graph with $\Delta\geq 6$ is a class 1 graph is $\Delta = 6$. We give a short proof of the following statement: there is no 6-critical plane graph $G$, such that every vertex of…
We settle a problem of Havel by showing that there exists an absolute constant d such that if G is a planar graph in which every two distinct triangles are at distance at least d, then G is 3-colorable. In fact, we prove a more general…
Two cycles are {\em adjacent} if they have an edge in common. Suppose that $G$ is a planar graph, for any two adjacent cycles $C_{1}$ and $C_{2}$, we have $|C_{1}| + |C_{2}| \geq 11$, in particular, when $|C_{1}| = 5$, $|C_{2}| \geq 7$. We…
A facial unique-maximum coloring of a plane graph is a proper vertex coloring by natural numbers where on each face $\alpha$ the maximal color appears exactly once on the vertices of $\alpha$. Fabrici and G\"oring proved that six colors are…
An $r$-uniform hypergraph is uniquely $k$-colorable if there exists exactly one partition of its vertex set into $k$ parts such that every edge contains at most one vertex from each part. For integers $k \ge r \ge 2$, let $\Phi_{k,r}$…
We consider cell colorings of drawings of graphs in the plane. Given a multi-graph $G$ together with a drawing $\Gamma(G)$ in the plane with only finitely many crossings, we define a cell $k$-coloring of $\Gamma(G)$ to be a coloring of the…
We say that a signed graph is $k$-critical if it is not $k$-colorable but every one of its proper subgraphs is $k$-colorable. Using the definition of colorability due to Naserasr, Wang, and Zhu that extends the notion of circular…