scholarly journals The majority coloring of the join and Cartesian product of some digraph

2022 ◽  
Vol 355 ◽  
pp. 02004
Author(s):  
Mei Shi ◽  
Weihao Xia ◽  
Mingyue Xiao ◽  
Jihui Wang
Keyword(s):  
Cartesian Product ◽  
Vertex Coloring ◽  
Join Graph

A majority coloring of a digraph is a vertex coloring such that for every vertex, the number of vertices with the same color in the out-neighborhood does not exceed half of its out-degree. Kreutzer, Oum, Seymour and van der Zyper proved that every digraph is majority 4-colorable and conjecture that every digraph has a majority 3-coloring. This paper mainly studies the majority coloring of the joint and Cartesian product of some special digraphs and proved the conjecture is true for the join graph and the Cartesian product. According to the influence of the number of vertices in digraph, we prove the majority coloring of the joint and Cartesian product of some digraph.

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.

Star chromatic number of some graphs

2021 ◽  
pp. 2150089
Author(s):  
S. Akbari ◽  
M. CHAVOOSHI ◽  
M. Ghanbari ◽  
S. Taghian
Keyword(s):  
Chromatic Number ◽  
Cartesian Product ◽  
Vertex Coloring ◽  
Star Coloring ◽  
Proper Vertex

A proper vertex coloring of a graph [Formula: see text] is called a star coloring if every two color classes induce a forest whose each component is a star, which means there is no bicolored [Formula: see text] in [Formula: see text]. In this paper, we show that the Cartesian product of any two cycles, except [Formula: see text] and [Formula: see text], has a [Formula: see text]-star coloring.

scholarly journals On the 1-2-3-conjecture

2015 ◽  
Vol Vol. 17 no. 1 (Graph Theory) ◽  
Author(s):  
Akbar Davoodi ◽  
Behnaz Omoomi
Keyword(s):  
Graph Theory ◽  
Cartesian Product ◽  
Sufficient Conditions ◽  
Bipartite Graphs ◽  
Vertex Coloring ◽  
Cartesian Product Of Graphs ◽  
Edge Weighting ◽  
International Audience ◽  
Product Of Graphs ◽  
Proper Vertex

Graph Theory International audience A k-edge-weighting of a graph G is a function w:E(G)→{1,…,k}. An edge-weighting naturally induces a vertex coloring c, where for every vertex v∈V(G), c(v)=∑e∼vw(e). If the induced coloring c is a proper vertex coloring, then w is called a vertex-coloring k-edge-weighting (VC k-EW). Karoński et al. (J. Combin. Theory Ser. B, 91 (2004) 151 13;157) conjectured that every graph admits a VC 3-EW. This conjecture is known as the 1-2-3-conjecture. In this paper, first, we study the vertex-coloring edge-weighting of the Cartesian product of graphs. We prove that if the 1-2-3-conjecture holds for two graphs G and H, then it also holds for G□H. Also we prove that the Cartesian product of connected bipartite graphs admits a VC 2-EW. Moreover, we present several sufficient conditions for a graph to admit a VC 2-EW. Finally, we explore some bipartite graphs which do not admit a VC 2-EW.

More on Single Valued Neutrosophic R-dynamic Vertex Coloring and R-dynamic Edge Coloring of graphs

2021 ◽  
pp. 16-27
Author(s):  
Aparna V. ◽  
◽  
Mohanapriya N. ◽  
Broumi Said ◽  
◽  
...  
Keyword(s):  
Cartesian Product ◽  
Edge Coloring ◽  
Vertex Coloring ◽  
Neutrosophic Sets

The notion of neutrosophic sets facilitates the analysis of values that are unclear or indeterminate. In this paper, we discuss the single-valued neutrosophic R-dynamic vertex coloring of the Cartesian product of SVNG’sand join of SVG's. Further, we have described the concept of single-valued neutrosophic R-dynamic edge coloring and provided some examples and theorems.

scholarly journals Sum List Coloring $2 \times n$ Arrays

10.37236/1669 ◽  
2002 ◽  
Vol 9 (1) ◽  
Author(s):  
Garth Isaak
Keyword(s):  
Cartesian Product ◽  
Edge Coloring ◽  
Vertex Coloring ◽  
List Coloring ◽  
Proper Coloring ◽  
List Edge Coloring ◽  
Choice Number ◽  
Sum Choice Number

A graph is $f$-choosable if for every collection of lists with list sizes specified by $f$ there is a proper coloring using colors from the lists. The sum choice number is the minimum over all choosable functions $f$ of the sum of the sizes in $f$. We show that the sum choice number of a $2 \times n$ array (equivalent to list edge coloring $K_{2,n}$ and to list vertex coloring the cartesian product $K_2 \square K_n$) is $n^2 + \lceil 5n/3 \rceil$.

CARTESIAN PRODUCT ON FUZZY IDEALS OF A TERNARY Γ-SEMIGROUP

2020 ◽  
Vol 9 (3) ◽  
pp. 1189-1195 ◽  
Author(s):  
Y. Bhargavi ◽  
T. Eswarlal ◽  
S. Ragamayi
Keyword(s):  
Cartesian Product

scholarly journals Indirect identification of horizontal gene transfer

2021 ◽  
Vol 83 (1) ◽  
Author(s):  
David Schaller ◽  
Manuel Lafond ◽  
Peter F. Stadler ◽  
Nicolas Wieseke ◽  
Marc Hellmuth
Keyword(s):  
Gene Transfer ◽  
Horizontal Gene Transfer ◽  
Polynomial Time ◽  
Divergence Time ◽  
Recognition Algorithm ◽  
Vertex Coloring ◽  
Implicit Methods ◽  
Colored Graphs ◽  
Wide Range ◽  
Complete Multipartite

AbstractSeveral implicit methods to infer horizontal gene transfer (HGT) focus on pairs of genes that have diverged only after the divergence of the two species in which the genes reside. This situation defines the edge set of a graph, the later-divergence-time (LDT) graph, whose vertices correspond to genes colored by their species. We investigate these graphs in the setting of relaxed scenarios, i.e., evolutionary scenarios that encompass all commonly used variants of duplication-transfer-loss scenarios in the literature. We characterize LDT graphs as a subclass of properly vertex-colored cographs, and provide a polynomial-time recognition algorithm as well as an algorithm to construct a relaxed scenario that explains a given LDT. An edge in an LDT graph implies that the two corresponding genes are separated by at least one HGT event. The converse is not true, however. We show that the complete xenology relation is described by an rs-Fitch graph, i.e., a complete multipartite graph satisfying constraints on the vertex coloring. This class of vertex-colored graphs is also recognizable in polynomial time. We finally address the question “how much information about all HGT events is contained in LDT graphs” with the help of simulations of evolutionary scenarios with a wide range of duplication, loss, and HGT events. In particular, we show that a simple greedy graph editing scheme can be used to efficiently detect HGT events that are implicitly contained in LDT graphs.

scholarly journals THE GENERAL POSITION NUMBER OF THE CARTESIAN PRODUCT OF TWO TREES

2020 ◽  
pp. 1-10
Author(s):  
JING TIAN ◽  
KEXIANG XU ◽  
SANDI KLAVŽAR
Keyword(s):  
Shortest Path ◽  
Connected Graph ◽  
General Position ◽  
Cartesian Product

Abstract The general position number of a connected graph is the cardinality of a largest set of vertices such that no three pairwise-distinct vertices from the set lie on a common shortest path. In this paper it is proved that the general position number is additive on the Cartesian product of two trees.

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