Acyclic Coloring with Few Division Vertices

2012 ◽  
pp. 86-99
Author(s):  
Debajyoti Mondal ◽  
Rahnuma Islam Nishat ◽  
Md. Saidur Rahman ◽  
Sue Whitesides
Keyword(s):  
Acyclic Coloring

Acyclic 4-choosability of planar graphs without intersecting short cycles

2018 ◽  
Vol 10 (01) ◽  
pp. 1850014
Author(s):  
Yingcai Sun ◽  
Min Chen ◽  
Dong Chen
Keyword(s):  
Planar Graphs ◽  
Vertex Coloring ◽  
Discrete Mathematics ◽  
Sufficient Condition ◽  
Acyclic Coloring ◽  
Proper Vertex

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.

Acyclic coloring of graphs with some girth restriction

2015 ◽  
Vol 31 (4) ◽  
pp. 1399-1404 ◽  
Author(s):  
Jiansheng Cai ◽  
Binlu Feng ◽  
Guiying Yan
Keyword(s):  
Acyclic Coloring

scholarly journals Acyclic coloring with few division vertices

2013 ◽  
Vol 23 ◽  
pp. 42-53 ◽  
Author(s):  
Debajyoti Mondal ◽  
Rahnuma Islam Nishat ◽  
Md. Saidur Rahman ◽  
Sue Whitesides
Keyword(s):  
Acyclic Coloring

Star Coloring and Acyclic Coloring of Locally Planar Graphs

2010 ◽  
Vol 24 (1) ◽  
pp. 56-71 ◽  
Author(s):  
Ken-ichi Kawarabayashi ◽  
Bojan Mohar
Keyword(s):  
Planar Graphs ◽  
Acyclic Coloring ◽  
Star Coloring

scholarly journals Acyclic Coloring of Graphs of Maximum Degree $\Delta$

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Guillaume Fertin ◽  
André Raspaud
Keyword(s):  
Chromatic Number ◽  
Upper Bound ◽  
Maximum Degree ◽  
Linear Time ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Acyclic Coloring ◽  
International Audience ◽  
Better Than

International audience An acyclic coloring of a graph $G$ is a coloring of its vertices such that: (i) no two neighbors in $G$ are assigned the same color and (ii) no bicolored cycle can exist in $G$. The acyclic chromatic number of $G$ is the least number of colors necessary to acyclically color $G$, and is denoted by $a(G)$. We show that any graph of maximum degree $\Delta$ has acyclic chromatic number at most $\frac{\Delta (\Delta -1) }{ 2}$ for any $\Delta \geq 5$, and we give an $O(n \Delta^2)$ algorithm to acyclically color any graph of maximum degree $\Delta$ with the above mentioned number of colors. This result is roughly two times better than the best general upper bound known so far, yielding $a(G) \leq \Delta (\Delta -1) +2$. By a deeper study of the case $\Delta =5$, we also show that any graph of maximum degree $5$ can be acyclically colored with at most $9$ colors, and give a linear time algorithm to achieve this bound.

scholarly journals Plurigraph Coloring and Scheduling Problems

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.

scholarly journals The Diachromatic Number of Digraphs

10.37236/7807 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Gabriela Araujo-Pardo ◽  
Juan José Montellano-Ballesteros ◽  
Mika Olsen ◽  
Christian Rubio-Montiel
Keyword(s):  
Chromatic Number ◽  
Directed Graphs ◽  
Interpolation Property ◽  
Acyclic Coloring ◽  
Chromatic Class

We consider the extension to directed graphs of the concept of achromatic number in terms of acyclic vertex colorings. The achromatic number have been intensely studied since it was introduced by Harary, Hedetniemi and Prins in 1967. The dichromaticnumber is a generalization of the chromatic number for digraphs defined by Neumann-Lara in 1982. A coloring of a digraph is an acyclic coloring if each subdigraph induced by each chromatic class is acyclic, and a coloring is complete if for any pair of chromatic classes $x,y$, there is an arc from $x$ to $y$ and an arc from $y$ to $x$. The dichromatic and diachromatic numbers are, respectively, the smallest and the largest number of colors in a complete acyclic coloring. We give some general results for the diachromatic number and study it for tournaments. We also show that the interpolation property for complete acyclic colorings does hold and establish Nordhaus-Gaddum relations.

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