Planar graphs with maximum degree 7 and without 5-cycles are 8-totally-colorable

It is proved that planar graphs with maximum degree 7 and without 5-cycles are 8-totally-colorable.     

Choosability of the square of planar subcubic graphs with large girth

We show that the choice number of the square of a subcubic graph with maximum average degree less than 18/7 is at most 6. As a corollary, we get that the choice number of the square of a subcubic planar graph with girth at least 9 is at most 6. We then show that the choice number of the square of a subcubic planar graph with girth at least 13 is at most 5.     

On the 9-total-colorability of planar graphs with maximum degree 8 and without intersecting triangles

Recently, Kowalik, Sereni, and ?krekovski proved that planar graphs with maximum degree 9 are 10-totally colorable. This work proves that planar graphs with maximum degree 8 and without intersecting triangles are 9-totally colorable.     

Covering projective planar graphs with three forests

It is known that all planar graphs and all projective planar graphs have an edge partition into three forests. Gonçalves proved that every planar graph has an edge partition into three forests, one having maximum degree at most four [5]. In this paper, we prove that every projective planar graph has an edge partition into three forests, one having maximum degree at most four.     

Subcubic planar graphs of girth 7 are class I

We prove that planar graphs of maximum degree 3 and of girth at least 7 are 3-edge-colorable, extending the previous result for girth at least 8 by Kronk, Radlowski, and Franen from 1974.     

Hard coloring problems in low degree planar bipartite graphs

In this paper we prove that the PRECOLORING EXTENSION problem on graphs of maximum degree 3 is polynomially solvable, but even its restricted version with 3 colors is NP-complete on planar bipartite graphs of maximum degree 4.The restricted version of LIST COLORING, in which the union of all lists consists of 3 colors, is shown to be NP-complete on planar 3-regular bipartite graphs.     

An Oriented 6-Coloring of Planar Graphs with Girth at Least 9

We prove that every oriented graph with a maximum average degree less than 18/7 admits a homomorphism into \(P_{7}^{*}\), the Paley tournament of order seven with one vertex deleted. In particular, every oriented planar graph of girth at least 9 has a homomorphism into \(P_{7}^{*}\), whence every planar graph of girth at least 9 has oriented chromatic number at most 6.     

Injective colorings of planar graphs with few colors

An injective coloring of a graph is a vertex coloring where two vertices have distinct colors if a path of length two exists between them. In this paper some results on injective colorings of planar graphs with few colors are presented. We show that all planar graphs of girth ≥ 19 and maximum degree Δ are injectively Δ-colorable. We also show that all planar graphs of girth ≥ 10 are injectively (Δ+1)-colorable, that Δ+4 colors are sufficient for planar graphs of girth ≥ 5 if Δ is large enough, and that subcubic planar graphs of girth ≥ 7 are injectively 5-colorable.     

Neighbor Sum Distinguishing Total Choice Number of Planar Graphs without 6-cycles

Pil?niak and Wo?niak put forward the concept of neighbor sum distinguishing (NSD) total coloring and conjectured that any graph with maximum degree Δ admits an NSD total (Δ+3)-coloring in 2015. In 2016, Qu et al. showed that the list version of the conjecture holds for any planar graph with Δ ≥ 13. In this paper, we prove that any planar graph with Δ ≥ 7 but without 6-cycles satisfies the list version of the conjecture.     

2-connected coverings of bounded degree in 3-connected graphs

In a recent paper, Barnette showed that every 3-connected planar graph has a 2-connected spanning subgraph of maximum degree at most fifteen, he also constructed a planar triangulation that does not have 2-connected spanning subgraphs of maximum degree five. In this paper, we show that every 3-connected graph which is embeddable in the sphere, the projective plane, the torus or the Klein bottle has a 2-connected spanning subgraph of maximum degree at most six. © 1995 John Wiley & Sons, Inc.