scholarly journals On the existence and non-existence of improper homomorphisms of oriented and $2$-edge-coloured graphs to reflexive targets

2021 ◽  
Vol vol. 23 no. 1 (Graph Theory) ◽  
Author(s):  
Christopher Duffy ◽  
Sonja Linghui Shan
Keyword(s):  
Chromatic Number ◽  
Open Problem ◽  
Planar Graphs ◽  
Oriented Graphs ◽  
Tree Width ◽  
Np Complete

We consider non-trivial homomorphisms to reflexive oriented graphs in which some pair of adjacent vertices have the same image. Using a notion of convexity for oriented graphs, we study those oriented graphs that do not admit such homomorphisms. We fully classify those oriented graphs with tree-width $2$ that do not admit such homomorphisms and show that it is NP-complete to decide if a graph admits an orientation that does not admit such homomorphisms. We prove analogous results for $2$-edge-coloured graphs. We apply our results on oriented graphs to provide a new tool in the study of chromatic number of orientations of planar graphs -- a long-standing open problem.

scholarly journals Consecutive Colouring of Oriented Graphs

2021 ◽  
Vol 76 (4) ◽  
Author(s):  
Marta Borowiecka-Olszewska ◽  
Ewa Drgas-Burchardt ◽  
Nahid Yelene Javier-Nol ◽  
Rita Zuazua
Keyword(s):  
Planar Graphs ◽  
Maximum Degree ◽  
Bipartite Graphs ◽  
Complete Graphs ◽  
Oriented Graphs ◽  
Complete Problem ◽  
Np Complete

AbstractWe consider arc colourings of oriented graphs such that for each vertex the colours of all out-arcs incident with the vertex and the colours of all in-arcs incident with the vertex form intervals. We prove that the existence of such a colouring is an NP-complete problem. We give the solution of the problem for r-regular oriented graphs, transitive tournaments, oriented graphs with small maximum degree, oriented graphs with small order and some other classes of oriented graphs. We state the conjecture that for each graph there exists a consecutive colourable orientation and confirm the conjecture for complete graphs, 2-degenerate graphs, planar graphs with girth at least 8, and bipartite graphs with arboricity at most two that include all planar bipartite graphs. Additionally, we prove that the conjecture is true for all perfect consecutively colourable graphs and for all forbidden graphs for the class of perfect consecutively colourable graphs.

scholarly journals Competitive Colorings of Oriented Graphs

10.37236/1611 ◽  
2000 ◽  
Vol 8 (2) ◽  
Author(s):  
H. A. Kierstead ◽  
W. T. Trotter
Keyword(s):  
Positive Integer ◽  
Chromatic Number ◽  
Complete Graph ◽  
Planar Graphs ◽  
General Technique ◽  
Oriented Graphs ◽  
Game Chromatic Number ◽  
Closed Class

Nešetřil and Sopena introduced a concept of oriented game chromatic number and developed a general technique for bounding this parameter. In this paper, we combine their technique with concepts introduced by several authors in a series of papers on game chromatic number to show that for every positive integer $k$, there exists an integer $t$ so that if ${\cal C}$ is a topologically closed class of graphs and ${\cal C}$ does not contain a complete graph on $k$ vertices, then whenever $G$ is an orientation of a graph from ${\cal C}$, the oriented game chromatic number of $G$ is at most $t$. In particular, oriented planar graphs have bounded oriented game chromatic number. This answers a question raised by Nešetřil and Sopena. We also answer a second question raised by Nešetřil and Sopena by constructing a family of oriented graphs for which oriented game chromatic number is bounded but extended Go number is not.

scholarly journals Negative results on acyclic improper colorings

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Pascal Ochem
Keyword(s):  
Chromatic Number ◽  
Planar Graphs ◽  
Maximum Degree ◽  
Np Completeness ◽  
Negative Results ◽  
International Audience ◽  
Np Complete ◽  
Large Girth

International audience Raspaud and Sopena showed that the oriented chromatic number of a graph with acyclic chromatic number $k$ is at most $k2^{k-1}$. We prove that this bound is tight for $k \geq 3$. We also show that some improper and/or acyclic colorings are $\mathrm{NP}$-complete on a class $\mathcal{C}$ of planar graphs. We try to get the most restrictive conditions on the class $\mathcal{C}$, such as having large girth and small maximum degree. In particular, we obtain the $\mathrm{NP}$-completeness of $3$-$\mathrm{ACYCLIC \space COLORABILITY}$ on bipartite planar graphs with maximum degree $4$, and of $4$-$\mathrm{ACYCLIC \space COLORABILITY}$ on bipartite planar graphs with maximum degree $8$.

r-Hued coloring of planar graphs without short cycles

2020 ◽  
Vol 12 (03) ◽  
pp. 2050034
Author(s):  
Yuehua Bu ◽  
Xiaofang Wang
Keyword(s):  
Planar Graph ◽  
Chromatic Number ◽  
Planar Graphs ◽  
Minimum Integer

A [Formula: see text]-hued coloring of a graph [Formula: see text] is a proper [Formula: see text]-coloring [Formula: see text] such that [Formula: see text] for any vertex [Formula: see text]. The [Formula: see text]-hued chromatic number of [Formula: see text], written [Formula: see text], is the minimum integer [Formula: see text] such that [Formula: see text] has a [Formula: see text]-hued coloring. In this paper, we show that [Formula: see text] if [Formula: see text] and [Formula: see text] is a planar graph without [Formula: see text]-cycles or if [Formula: see text] is a planar graph without [Formula: see text]-cycles and no [Formula: see text]-cycle is intersect with [Formula: see text]-cycles, [Formula: see text], then [Formula: see text], where [Formula: see text].

An improved upper bound for the acyclic chromatic number of 1-planar graphs

2020 ◽  
Vol 283 ◽  
pp. 275-291
Author(s):  
Wanshun Yang ◽  
Weifan Wang ◽  
Yiqiao Wang
Keyword(s):  
Chromatic Number ◽  
Upper Bound ◽  
Planar Graphs

k-Degenerate Graphs

1970 ◽  
Vol 22 (5) ◽  
pp. 1082-1096 ◽  
Author(s):  
Don R. Lick ◽  
Arthur T. White
Keyword(s):  
Bipartite Graph ◽  
Chromatic Number ◽  
Complete Graph ◽  
Planar Graphs ◽  
Complete Bipartite Graph ◽  
Natural Numbers ◽  
Image Position ◽  
Disconnected Graphs ◽  
Totally Disconnected

Graphs possessing a certain property are often characterized in terms of a type of configuration or subgraph which they cannot possess. For example, a graph is totally disconnected (or, has chromatic number one) if and only if it contains no lines; a graph is a forest (or, has point-arboricity one) if and only if it contains no cycles. Chartrand, Geller, and Hedetniemi [2] defined a graph to have property Pn if it contains no subgraph homeomorphic from the complete graph Kn+1 or the complete bipartite graphFor the first four natural numbers n, the graphs with property Pn are exactly the totally disconnected graphs, forests, outerplanar and planar graphs, respectively. This unification suggested the extension of many results known to hold for one of the above four classes of graphs to one or more of the remaining classes.

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