Star Coloring and Acyclic Coloring of Locally Planar Graphs

A proper vertex coloring of [Formula: see text] is acyclic if [Formula: see text] contains no bicolored cycle. Namely, every cycle of [Formula: see text] must be colored with at least three colors. [Formula: see text] is acyclically [Formula: see text]-colorable if for a given list assignment [Formula: see text], there exists an acyclic coloring [Formula: see text] of [Formula: see text] such that [Formula: see text] for all [Formula: see text]. If [Formula: see text] is acyclically [Formula: see text]-colorable for any list assignment with [Formula: see text] for all [Formula: see text], then [Formula: see text] is acyclically [Formula: see text]-choosable. In this paper, we prove that planar graphs without intersecting [Formula: see text]-cycles are acyclically [Formula: see text]-choosable. This provides a sufficient condition for planar graphs to be acyclically 4-choosable and also strengthens a result in [M. Montassier, A. Raspaud and W. Wang, Acyclic 4-choosability of planar graphs without cycles of specific lengths, in Topics in Discrete Mathematics, Algorithms and Combinatorics, Vol. 26 (Springer, Berlin, 2006), pp. 473–491] which says that planar graphs without [Formula: see text]-, [Formula: see text]-cycles and intersecting 3-cycles are acyclically 4-choosable.

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Star coloring planar graphs from small lists

Journal of Graph Theory ◽

10.1002/jgt.20431 ◽

2009 ◽

pp. n/a-n/a

Author(s):

André Kündgen ◽

Craig Timmons

Keyword(s):

Planar Graphs ◽

Star Coloring

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Plurigraph Coloring and Scheduling Problems

The Electronic Journal of Combinatorics ◽

10.37236/6818 ◽

2017 ◽

Vol 24 (2) ◽

Author(s):

John Machacek

Keyword(s):

Symmetric Function ◽

Degree Sequence ◽

Simplicial Complexes ◽

Vertex Coloring ◽

Scheduling Problems ◽

Acyclic Coloring ◽

New Type ◽

Star Coloring ◽

Chromatic Symmetric Function ◽

Special Case

We define a new type of vertex coloring which generalizes vertex coloring in graphs, hypergraphs, and simplicial complexes. This coloring also generalizes oriented coloring, acyclic coloring, and star coloring. There is an associated symmetric function in noncommuting variables for which we give a deletion-contraction formula. In the case of graphs this symmetric function in noncommuting variables agrees with the chromatic symmetric function in noncommuting variables of Gebhard and Sagan. Our vertex coloring is a special case of the scheduling problems defined by Breuer and Klivans. We show how the deletion-contraction law can be applied to scheduling problems. Also, we show that the chromatic symmetric function determines the degree sequence of uniform hypertrees, but there exists pairs of 3-uniform hypertrees which are not isomorphic yet have the same chromatic symmetric function.

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Coloring with no $2$-Colored $P_4$'s

The Electronic Journal of Combinatorics ◽

10.37236/1779 ◽

2004 ◽

Vol 11 (1) ◽

Cited By ~ 58

Author(s):

Michael O. Albertson ◽

Glenn G. Chappell ◽

H. A. Kierstead ◽

André Kündgen ◽

Radhika Ramamurthi

Keyword(s):

Planar Graph ◽

Planar Graphs ◽

Short Proof ◽

Cubic Graphs ◽

Proper Coloring ◽

Be Star ◽

Tree Width ◽

Star Coloring

A proper coloring of the vertices of a graph is called a star coloring if every two color classes induce a star forest. Star colorings are a strengthening of acyclic colorings, i.e., proper colorings in which every two color classes induce a forest. We show that every acyclic $k$-coloring can be refined to a star coloring with at most $(2k^2-k)$ colors. Similarly, we prove that planar graphs have star colorings with at most 20 colors and we exhibit a planar graph which requires 10 colors. We prove several other structural and topological results for star colorings, such as: cubic graphs are $7$-colorable, and planar graphs of girth at least $7$ are $9$-colorable. We provide a short proof of the result of Fertin, Raspaud, and Reed that graphs with tree-width $t$ can be star colored with ${t+2\choose2}$ colors, and we show that this is best possible.

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Star coloring bipartite planar graphs

Journal of Graph Theory ◽

10.1002/jgt.20342 ◽

2009 ◽

Vol 60 (1) ◽

pp. 1-10 ◽

Cited By ~ 13

Author(s):

H. A. Kierstead ◽

André Kündgen ◽

Craig Timmons

Keyword(s):

Planar Graphs ◽

Star Coloring

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Acyclic coloring of IC-planar graphs

Discrete Mathematics ◽

10.1016/j.disc.2019.111623 ◽

2019 ◽

Vol 342 (12) ◽

pp. 111623 ◽

Cited By ~ 3

Author(s):

Wanshun Yang ◽

Weifan Wang ◽

Yiqiao Wang

Keyword(s):

Planar Graphs ◽

Acyclic Coloring

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Star Coloring High Girth Planar Graphs

The Electronic Journal of Combinatorics ◽

10.37236/848 ◽

2008 ◽

Vol 15 (1) ◽

Cited By ~ 11

Author(s):

Craig Timmons

Keyword(s):

Planar Graph ◽

Planar Graphs ◽

Proper Coloring ◽

Be Star ◽

Star Coloring

A star coloring of a graph is a proper coloring such that no path on four vertices is 2-colored. We prove that every planar graph with girth at least 9 can be star colored using 5 colors, and that every planar graph with girth at least 14 can be star colored using 4 colors; the figure 4 is best possible. We give an example of a girth 7 planar graph that requires 5 colors to star color.

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8-star-choosability of a graph with maximum average degree less than 3

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.561 ◽

2011 ◽

Vol Vol. 13 no. 3 (Graph and Algorithms) ◽

Author(s):

Min Chen ◽

André Raspaud ◽

Weifan Wang

Keyword(s):

Graph Theory ◽

Chromatic Number ◽

Planar Graphs ◽

Average Degree ◽

Vertex Coloring ◽

Maximum Average Degree ◽

International Audience ◽

Star Coloring ◽

List Chromatic Number ◽

Proper Vertex

Graphs and Algorithms International audience A proper vertex coloring of a graphGis called a star-coloring if there is no path on four vertices assigned to two colors. The graph G is L-star-colorable if for a given list assignment L there is a star-coloring c such that c(v) epsilon L(v). If G is L-star-colorable for any list assignment L with vertical bar L(v)vertical bar \textgreater= k for all v epsilon V(G), then G is called k-star-choosable. The star list chromatic number of G, denoted by X-s(l)(G), is the smallest integer k such that G is k-star-choosable. In this article, we prove that every graph G with maximum average degree less than 3 is 8-star-choosable. This extends a result that planar graphs of girth at least 6 are 8-star-choosable [A. Kundgen, C. Timmons, Star coloring planar graphs from small lists, J. Graph Theory, 63(4): 324-337, 2010].

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