On 4-critical planar graphs with high edge density

The edge-density of 4-critical planar graphs

On the edge-density of 4-critical graphs

Gallai conjectured that every 4-critical graph on n vertices has at least 53n − 23 edges. We prove this conjecture for 4-critical graphs in which the subgraph induced by vertices of degree 3 is connected.

Planar 4-critical graphs with four triangles

By the Grünbaum-Aksenov Theorem (extending Grötzsch’s Theorem) every planar graph with at most three triangles is 3-colorable. However, there are infinitely many planar 4-critical graphs with exactly four triangles. We describe all such graphs. This answers a question of Erdős from 1990.

Drawing some 4-regular planar graphs with integer edge lengths

A classic result of Fáry states that every planar graph can be drawn in the plane without crossings using only straight line segments. Harborth et al. conjecture that every planar graph has such a drawing where every edge length is integral. Biedl proves that every planar graph of maximum degree 4 that is not 4-regular has such a straight-line embedding, but the techniques are insufficient for ...

The Size of Edge-critical Uniquely 3-Colorable Planar Graphs

A graph G is 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 graph for any edge e ∈ E(G). In this paper, we prove that if G is an edge-critical uniquely 3-colorable planar graph, then |E(G)| 6 83 |V (G)| − 17 3 . On the other hand, there exis...