scholarly journals Locally Identifying Coloring of Graphs

10.37236/2417 ◽  
2012 ◽  
Vol 19 (2) ◽  
Author(s):  
Louis Esperet ◽  
Sylvain Gravier ◽  
Mickaël Montassier ◽  
Pascal Ochem ◽  
Aline Parreau
Keyword(s):  
Bipartite Graph ◽  
Planar Graphs ◽  
Maximum Degree ◽  
Perfect Graphs ◽  
Vertex Coloring ◽  
Complete Problem ◽  
Minimum Number ◽  
Np Complete ◽  
Large Girth ◽  
Proper Vertex

We introduce the notion of locally identifying coloring of a graph. A proper vertex-coloring $c$ of a graph $G$ is said to be locally identifying, if for any adjacent vertices $u$ and $v$ with distinct closed neighborhoods, the sets of colors that appear in the closed neighborhood of $u$ and $v$, respectively, are distinct. Let $\chi_{\rm{lid}}(G)$ be the minimum number of colors used in a locally identifying vertex-coloring of $G$. In this paper, we give several bounds on $\chi_{\rm{lid}}$ for different families of graphs (planar graphs, some subclasses of perfect graphs, graphs with bounded maximum degree) and prove that deciding whether $\chi_{\rm{lid}}(G)=3$ for a subcubic bipartite graph $G$ with large girth is an NP-complete problem.

scholarly journals Linear choosability of graphs

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Louis Esperet ◽  
Mickael Montassier ◽  
André Raspaud
Keyword(s):  
Planar Graphs ◽  
Maximum Degree ◽  
Oriented Graph ◽  
Vertex Coloring ◽  
Proper Coloring ◽  
Maximum Average Degree ◽  
Complete Problem ◽  
International Audience ◽  
Np Complete ◽  
Proper Vertex

International audience A proper vertex coloring of a non oriented graph $G=(V,E)$ is linear if the graph induced by the vertices of two color classes is a forest of paths. A graph $G$ is $L$-list colorable if for a given list assignment $L=\{L(v): v∈V\}$, there exists a proper coloring $c$ of $G$ such that $c(v)∈L(v)$ for all $v∈V$. If $G$ is $L$-list colorable for every list assignment with $|L(v)|≥k$ for all $v∈V$, then $G$ is said $k$-choosable. A graph is said to be lineary $k$-choosable if the coloring obtained is linear. In this paper, we investigate the linear choosability of graphs for some families of graphs: graphs with small maximum degree, with given maximum average degree, planar graphs... Moreover, we prove that determining whether a bipartite subcubic planar graph is lineary 3-colorable is an NP-complete 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 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 Clustered 3-colouring graphs of bounded degree

2021 ◽  
pp. 1-13
Author(s):  
Vida Dujmović ◽  
Louis Esperet ◽  
Pat Morin ◽  
Bartosz Walczak ◽  
David R. Wood
Keyword(s):  
Planar Graphs ◽  
Maximum Degree ◽  
Bounded Degree ◽  
Bounded Number ◽  
Proper Vertex ◽  
Vertex Colouring

Abstract A (not necessarily proper) vertex colouring of a graph has clustering c if every monochromatic component has at most c vertices. We prove that planar graphs with maximum degree $\Delta$ are 3-colourable with clustering $O(\Delta^2)$ . The previous best bound was $O(\Delta^{37})$ . This result for planar graphs generalises to graphs that can be drawn on a surface of bounded Euler genus with a bounded number of crossings per edge. We then prove that graphs with maximum degree $\Delta$ that exclude a fixed minor are 3-colourable with clustering $O(\Delta^5)$ . The best previous bound for this result was exponential in $\Delta$ .

On the double total dominator chromatic number of graphs

2021 ◽  
pp. 2150154
Author(s):  
Fairouz Beggas ◽  
Hamamache Kheddouci ◽  
Walid Marweni
Keyword(s):  
Chromatic Number ◽  
Minimum Degree ◽  
Domination Number ◽  
Vertex Coloring ◽  
Total Domination Number ◽  
Close Relationship ◽  
Minimum Number ◽  
Number Formula ◽  
Dominator Coloring ◽  
Proper Vertex

In this paper, we introduce and study a new coloring problem of graphs called the double total dominator coloring. A double total dominator coloring of a graph [Formula: see text] with minimum degree at least 2 is a proper vertex coloring of [Formula: see text] such that each vertex has to dominate at least two color classes. The minimum number of colors among all double total dominator coloring of [Formula: see text] is called the double total dominator chromatic number, denoted by [Formula: see text]. Therefore, we establish the close relationship between the double total dominator chromatic number [Formula: see text] and the double total domination number [Formula: see text]. We prove the NP-completeness of the problem. We also examine the effects on [Formula: see text] when [Formula: see text] is modified by some operations. Finally, we discuss the [Formula: see text] number of square of trees by giving some bounds.

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.

scholarly journals Local antimagic vertex coloring of unicyclic graphs

2018 ◽  
Vol 2 (1) ◽  
pp. 30 ◽  
Author(s):  
Nuris Hisan Nazula ◽  
S Slamin ◽  
D Dafik
Keyword(s):  
Chromatic Number ◽  
Vertex Coloring ◽  
Unicyclic Graphs ◽  
Antimagic Labeling ◽  
Minimum Number ◽  
Proper Vertex

The local antimagic labeling on a graph G with |V| vertices and |E| edges is defined to be an assignment f : E --&gt; {1, 2,..., |E|} so that the weights of any two adjacent vertices u and v are distinct, that is, w(u)̸  ̸= w(v) where w(u) = Σe∈<sub>E(u)</sub> f(e) and E(u) is the set of edges incident to u. Therefore, any local antimagic labeling induces a proper vertex coloring of G where the vertex u is assigned the color w(u). The local antimagic chromatic number, denoted by χla(G), is the minimum number of colors taken over all colorings induced by local antimagic labelings of G. In this paper, we present the local antimagic chromatic number of unicyclic graphs that is the graphs containing exactly one cycle such as kite and cycle with two neighbour pendants.

scholarly journals Acyclic colourings of graphs with bounded degree

2010 ◽  
Vol Vol. 12 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Mieczyslaw Borowiecki ◽  
Anna Fiedorowicz ◽  
Katarzyna Jesse-Jozefczyk ◽  
Elzbieta Sidorowicz
Keyword(s):  
Polynomial Time ◽  
Maximum Degree ◽  
Cubic Graph ◽  
Bounded Degree ◽  
Complete Problem ◽  
International Audience ◽  
Np Complete

Graphs and Algorithms International audience A k-colouring of a graph G is called acyclic if for every two distinct colours i and j, the subgraph induced in G by all the edges linking a vertex coloured with i and a vertex coloured with j is acyclic. In other words, there are no bichromatic alternating cycles. In 1999 Boiron et al. conjectured that a graph G with maximum degree at most 3 has an acyclic 2-colouring such that the set of vertices in each colour induces a subgraph with maximum degree at most 2. In this paper we prove this conjecture and show that such a colouring of a cubic graph can be determined in polynomial time. We also prove that it is an NP-complete problem to decide if a graph with maximum degree 4 has the above mentioned colouring.

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 The Local Antimagic Chromatic Numbers of Some Join Graphs

2021 ◽  
Vol 26 (4) ◽  
pp. 80
Author(s):  
Xue Yang ◽  
Hong Bian ◽  
Haizheng Yu ◽  
Dandan Liu
Keyword(s):  
Partial Solution ◽  
Vertex Coloring ◽  
Chromatic Numbers ◽  
Infinite Class ◽  
Antimagic Labeling ◽  
Join Graph ◽  
Complement Graph ◽  
Minimum Number ◽  
Vertex Label ◽  
Proper Vertex

Let G=(V(G),E(G)) be a connected graph with n vertices and m edges. A bijection f:E(G)→{1,2,⋯,m} is an edge labeling of G. For any vertex x of G, we define ω(x)=∑e∈E(x)f(e) as the vertex label or weight of x, where E(x) is the set of edges incident to x, and f is called a local antimagic labeling of G, if ω(u)≠ω(v) for any two adjacent vertices u,v∈V(G). It is clear that any local antimagic labelling of G induces a proper vertex coloring of G by assigning the vertex label ω(x) to any vertex x of G. The local antimagic chromatic number of G, denoted by χla(G), is the minimum number of different vertex labels taken over all colorings induced by local antimagic labelings of G. In this paper, we present explicit local antimagic chromatic numbers of Fn∨K2¯ and Fn−v, where Fn is the friendship graph with n triangles and v is any vertex of Fn. Moreover, we explicitly construct an infinite class of connected graphs G such that χla(G)=χla(G∨K2¯), where G∨K2¯ is the join graph of G and the complement graph of complete graph K2. This fact leads to a counterexample to a theorem of Arumugam et al. in 2017, and our result also provides a partial solution to Problem 3.19 in Lau et al. in 2021.

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