Complete colorings of planar graphs (original) (raw)

The Two-Coloring Number and Degenerate Colorings of Planar Graphs

Siam Journal on Discrete Mathematics, 2009

The two-coloring number of graphs, which was originally introduced in the study of the game chromatic number, also gives an upper bound on the degenerate chromatic number as introduced by Borodin. It is proved that the two-coloring number of any planar graph is at most nine. As a consequence, the degenerate list chromatic number of any planar graph is at most nine. It is also shown that the degenerate diagonal chromatic number is at most eleven and the degenerate diagonal list chromatic number is at most tweleve for all planar graphs.

List improper colorings of planar graphs with prescribed girth

Discrete Mathematics, 2000

A graph G is m-choosable with impropriety d, or simply (m; d) *-choosable, if for every list assignment L, where |L(v)|¿m for every v ∈ V (G), there exists an L-coloring of G such that every vertex of G has at most d neighbors colored with the same color as itself. Denote by g d the smallest number such that every planar graph of girth at least g d is (2; d) *-choosable. In this paper it is shown that g169, g267, g366 and g d = 5 for every d¿4.

Coloring Vertices and Faces of Locally Planar Graphs

Graphs and Combinatorics, 2006

If G is an embedded graph, a vertex-face r-coloring is a mapping that assigns a color from the set {1,. .. , r} to every vertex and every face of G such that different colors are assigned whenever two elements are either adjacent or incident. Let χ vf (G) denote the minimum r such that G has a vertex-face r-coloring. Ringel conjectured that if G is planar, then χ vf (G) ≤ 6. A graph G drawn on a surface S is said to be 1-embedded in S if every edge crosses at most one other edge. Borodin proved that if G is 1-embedded in the plane, then χ(G) ≤ 6. This result implies Ringel's conjecture. Ringel also stated a Heawood style theorem for 1-embedded graphs. We prove a slight strengthening of this result. If G is 1-embedded in S, let w(G) denote the edge-width of G, i.e. the length of a shortest non-contractible cycle in G. We show that if G is 1-embedded in S and w(G) is large enough, then the list chromatic number ch(G) is at most 8.

On the Three Colorability of Planar Graphs

In this paper we have given a new three colorability criteria for planar graphs that can be considered as an generalization of the Heawood and the Grotszch theorems with respect to the triangulation and cycles of length greater than ≥ 4. We have shown that an triangulated planar graph with k disjoint holes is 3-colorable if and only if every hole satises the parity symmetric property, where a hole is a cycle (face boundary) of length greater than 3 or an induced outerplanar subgraph.

Star Coloring and Acyclic Coloring of Locally Planar Graphs

SIAM Journal on Discrete Mathematics, 2010

It is proved that every graph embedded in a fixed surface with sufficiently large edge-width is acyclically 7-colorable and that its star chromatic number is at most 2s * 0 + 3, where s * 0 ≤ 20 is the maximum star chromatic number for the class of all planar graphs.

Acyclic edge coloring of planar graphs with colors

Discrete Applied Mathematics, 2012

An acyclic edge coloring of a graph is a proper edge coloring without bichromatic cycles. In 1978, it was conjectured that ∆(G) + 2 colors suffice for an acyclic edge coloring of every graph G [6]. The conjecture has been verified for several classes of graphs, however, the best known upper bound for as special class as planar graphs are, is ∆ + 12 [2]. In this paper, we study simple planar graphs which need only ∆(G) colors for an acyclic edge coloring. We show that a planar graph with girth g and maximum degree ∆ admits such acyclic edge coloring if g ≥ 12, or g ≥ 8 and ∆ ≥ 4, or g ≥ 7 and ∆ ≥ 5, or g ≥ 6 and ∆ ≥ 6, or g ≥ 5 and ∆ ≥ 10. Our results improve some previously known bounds.

Injective coloring of planar graphs

Discrete Applied Mathematics, 2009

A coloring of a graph G is injective if its restriction to the neighborhood of any vertex is injective. The injective chromatic number χ i (G) of a graph G is the least k such that there is an injective k-coloring. In this paper we prove that if G is a planar graph with girth g and maximum degree ∆, then (1) χ i (G) = ∆ if either g ≥ 20 and ∆ ≥ 3, or g ≥ 7 and ∆ ≥ 71; (2) χ i (G) ≤ ∆ + 1 if g ≥ 11; (3) χ i (G) ≤ ∆ + 2 if g ≥ 8.

Strong edge-coloring of planar graphs

Discrete Mathematics, 2014

A strong edge coloring of a graph is a proper edge coloring where the edges at distance at most two receive distinct colors. It is known that every planar graph with maximum degree ∆ has a strong edge coloring with at most 4 ∆ + 4 colors. We show that 3 ∆ + 6 colors suffice if the graph has girth 6, and 3 ∆ colors suffice if the girth is at least 7. Moreover, we show that cubic planar graphs with girth at least 6 can be strongly edge colored with at most 9 colors.

Planar graphs with maximum degree 4 are strongly 19-edge-colorable

Discrete Mathematics, 2018

A strong edge-coloring of a graph is a proper edge-coloring such that edges at distance at most 2 receive different colors. It is known that every planar graph has a strong edgecoloring by using at most 4∆ + 4 colors, where ∆ denotes the maximum degree of the graph. In this paper, we will show that 19 colors are enough to color a planar graph with maximum degree 4.