scholarly journals Near-Colorings: Non-Colorable Graphs and NP-Completeness

10.37236/3509 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
M. Montassier ◽  
P. Ochem
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Maximum Degree ◽  
Fixed Integer ◽  
Np Completeness ◽  
Vertex Set ◽  
Np Complete

A graph $G$ is $(d_1,...,d_l)$-colorable if the vertex set of $G$ can be partitioned into subsets $V_1,\ldots ,V_l$ such that the graph $G[V_i]$ induced by the vertices of $V_i$ has maximum degree at most $d_i$ for all $1 \leq i \leq l$. In this paper, we focus on complexity aspects of such colorings when $l=2,3$. More precisely, we prove that, for any fixed integers $k,j,g$ with $(k,j) \neq (0,0)$ and $g\geq3$, either every planar graph with girth at least $g$ is $(k,j)$-colorable or it is NP-complete to determine whether a planar graph with girth at least $g$ is $(k,j)$-colorable. Also, for any fixed integer $k$, it is NP-complete to determine whether a planar graph that is either $(0,0,0)$-colorable or non-$(k,k,1)$-colorable is $(0,0,0)$-colorable. Additionally, we exhibit non-$(3,1)$-colorable planar graphs with girth 5 and non-$(2,0)$-colorable planar graphs with girth 7. 

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 $8$.

scholarly journals Planar Graphs of Maximum Degree Six without 7-cycles are Class One

10.37236/2589 ◽  
2012 ◽  
Vol 19 (3) ◽  
Author(s):  
Danjun Huang ◽  
Weifan Wang
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Maximum Degree

In this paper, we prove that every planar graph of maximum degree six without 7-cycles is class one.

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.

A note on the 2-rainbow bondage numbers in graphs

2016 ◽  
Vol 09 (01) ◽  
pp. 1650013
Author(s):  
L. Asgharsharghi ◽  
S. M. Sheikholeslami ◽  
L. Volkmann
Keyword(s):  
Planar Graph ◽  
Maximum Degree ◽  
Minimum Weight ◽  
Domination Number ◽  
Minimum Cardinality ◽  
Bondage Number ◽  
Rainbow Domination ◽  
Vertex Set ◽  
Number Formula ◽  
Appl Math

A 2-rainbow dominating function (2RDF) of a graph [Formula: see text] is a function [Formula: see text] from the vertex set [Formula: see text] to the set of all subsets of the set [Formula: see text] such that for any vertex [Formula: see text] with [Formula: see text], the condition [Formula: see text] is fulfilled. The weight of a 2RDF [Formula: see text] is the value [Formula: see text]. The [Formula: see text]-rainbow domination number of a graph [Formula: see text], denoted by [Formula: see text], is the minimum weight of a 2RDF of [Formula: see text]. The rainbow bondage number [Formula: see text] of a graph [Formula: see text] with maximum degree at least two is the minimum cardinality of all sets [Formula: see text] for which [Formula: see text]. Dehgardi, Sheikholeslami and Volkmann, [The [Formula: see text]-rainbow bondage number of a graph, Discrete Appl. Math. 174 (2014) 133–139] proved that the rainbow bondage number of a planar graph does not exceed 15. In this paper, we generalize their result for graphs which admit a [Formula: see text]-cell embedding on a surface with non-negative Euler characteristic.

scholarly journals A Sufficient Condition for Planar Graphs of Maximum Degree 6 to be Totally 7-Colorable

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Enqiang Zhu ◽  
Yongsheng Rao
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Maximum Degree ◽  
Simple Graph ◽  
Total Coloring ◽  
Sufficient Condition ◽  
Open Case ◽  
Total Coloring Conjecture

A total k-coloring of a graph is an assignment of k colors to its vertices and edges such that no two adjacent or incident elements receive the same color. The total coloring conjecture (TCC) states that every simple graph G has a total ΔG+2-coloring, where ΔG is the maximum degree of G. This conjecture has been confirmed for planar graphs with maximum degree at least 7 or at most 5, i.e., the only open case of TCC is that of maximum degree 6. It is known that every planar graph G of ΔG≥9 or ΔG∈7,8 with some restrictions has a total ΔG+1-coloring. In particular, in (Shen and Wang, 2009), the authors proved that every planar graph with maximum degree 6 and without 4-cycles has a total 7-coloring. In this paper, we improve this result by showing that every diamond-free and house-free planar graph of maximum degree 6 is totally 7-colorable if every 6-vertex is not incident with two adjacent four cycles or three cycles of size p,q,ℓ for some p,q,ℓ∈3,4,4,3,3,4.

scholarly journals Colouring Planar Graphs With Three Colours and No Large Monochromatic Components

2014 ◽  
Vol 23 (4) ◽  
pp. 551-570 ◽  
Author(s):  
LOUIS ESPERET ◽  
GWENAËL JORET
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Maximum Degree

We prove the existence of a function $f :\mathbb{N} \to \mathbb{N}$ such that the vertices of every planar graph with maximum degree Δ can be 3-coloured in such a way that each monochromatic component has at most f(Δ) vertices. This is best possible (the number of colours cannot be reduced and the dependence on the maximum degree cannot be avoided) and answers a question raised by Kleinberg, Motwani, Raghavan and Venkatasubramanian in 1997. Our result extends to graphs of bounded genus.

scholarly journals Defective 3-Paintability of Planar Graphs

10.37236/7084 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Grzegorz Gutowski ◽  
Ming Han ◽  
Tomasz Krawczyk ◽  
Xuding Zhu
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Maximum Degree ◽  
Winning Strategy ◽  
Marked Vertex

A $d$-defective $k$-painting game on a graph $G$ is played by two players: Lister and Painter. Initially, each vertex is uncolored and has $k$ tokens. In each round, Lister marks a chosen set $M$ of uncolored vertices and removes one token from each marked vertex. In response, Painter colors vertices in a subset $X$ of $M$ which induce a subgraph $G[X]$ of maximum degree at most $d$. Lister wins the game if at the end of some round there is an uncolored vertex that has no more tokens left. Otherwise, all vertices eventually get colored and Painter wins the game. We say that $G$ is $d$-defective $k$-paintable if Painter has a winning strategy in this game. In this paper we show that every planar graph is 3-defective 3-paintable and give a construction of a planar graph that is not 2-defective 3-paintable.

scholarly journals An Update on Reconfiguring $10$-Colorings of Planar Graphs

10.37236/9391 ◽  
2020 ◽  
Vol 27 (4) ◽  
Author(s):  
Zdeněk Dvořák ◽  
Carl Feghali
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Vertex Set

The reconfiguration graph $R_k(G)$ for the $k$-colorings of a graph~$G$ has as vertex set the set of all possible proper $k$-colorings of $G$ and two colorings are adjacent if they differ in the color of exactly one vertex. A result of Bousquet and Perarnau (2016) regarding graphs of bounded degeneracy implies that if $G$ is a planar graph with $n$ vertices, then $R_{12}(G)$ has diameter at most $6n$. We improve on the number of colors, showing that $R_{10}(G)$ has diameter at most $8n$ for every planar graph $G$ with $n$ vertices.

scholarly journals Partitioning planar graphs with girth at least $ 9 $ into an edgeless graph and a graph with bounded size components

2021 ◽  
Vol 1 (3) ◽  
pp. 136-144
Author(s):  
Chunyu Tian ◽  
◽  
Lei Sun
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Vertex Set ◽  
Bounded Size

<abstract><p>In this paper, we study the problem of partitioning the vertex set of a planar graph with girth restriction into parts, also referred to as color classes, such that each part induces a graph with components of bounded order. An ($ \mathcal{I} $, $ \mathcal{O}_{k} $)-partition of a graph $ G $ is the partition of $ V(G) $ into two non-empty subsets $ V_{1} $ and $ V_{2} $, such that $ G[V_{1}] $ is an edgeless graph and $ G[V_{2}] $ is a graph with components of order at most $ k $. We prove that every planar graph with girth 9 and without intersecting $ 9 $-face admits an ($ \mathcal{I} $, $ \mathcal{O}_{6} $)-partition. This improves a result of Choi, Dross and Ochem (2020) which says every planar graph with girth at least $ 9 $ admits an ($ \mathcal{I} $, $ \mathcal{O}_{9} $)-partition.</p></abstract>

scholarly journals Vertex-Partitioning into Fixed Additive Induced-Hereditary Properties is NP-hard

10.37236/1799 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Alastair Farrugia
Keyword(s):  
Planar Graph ◽  
Line Graph ◽  
Perfect Graph ◽  
Np Hard ◽  
Arbitrary Graph ◽  
Np Completeness ◽  
Graph Properties ◽  
Special Cases ◽  
Np Complete ◽  
Hereditary Properties

Can the vertices of an arbitrary graph $G$ be partitioned into $A \cup B$, so that $G[A]$ is a line-graph and $G[B]$ is a forest? Can $G$ be partitioned into a planar graph and a perfect graph? The NP-completeness of these problems are special cases of our result: if ${\cal P}$ and ${\cal Q}$ are additive induced-hereditary graph properties, then $({\cal P}, {\cal Q})$-colouring is NP-hard, with the sole exception of graph $2$-colouring (the case where both ${\cal P}$ and ${\cal Q}$ are the set ${\cal O}$ of finite edgeless graphs). Moreover, $({\cal P}, {\cal Q})$-colouring is NP-complete iff ${\cal P}$- and ${\cal Q}$-recognition are both in NP. This completes the proof of a conjecture of Kratochvíl and Schiermeyer, various authors having already settled many sub-cases.

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