Local Antimagic Chromatic Number for Copies of Graphs

An edge labeling of a graph G=(V,E) using every label from the set {1,2,⋯,|E(G)|} exactly once is a local antimagic labeling if the vertex-weights are distinct for every pair of neighboring vertices, where a vertex-weight is the sum of labels of all edges incident with that vertex. Any local antimagic labeling induces a proper vertex coloring of G where the color of a vertex is its vertex-weight. This naturally leads to the concept of a local antimagic chromatic number. The local antimagic chromatic number is defined to be the minimum number of colors taken over all colorings of G induced by local antimagic labelings of G. In this paper, we estimate the bounds of the local antimagic chromatic number for disjoint union of multiple copies of a graph.

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Local antimagic vertex coloring of unicyclic graphs

Indonesian Journal of Combinatorics ◽

10.19184/ijc.2018.2.1.4 ◽

2018 ◽

Vol 2 (1) ◽

pp. 30 ◽

Cited By ~ 3

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 --> {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∈E(u) 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.

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On the double total dominator chromatic number of graphs

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830921501548 ◽

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.

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The Local Antimagic Chromatic Numbers of Some Join Graphs

Mathematical and Computational Applications ◽

10.3390/mca26040080 ◽

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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On local antimagic vertex coloring of corona products related to friendship and fan graph

Indonesian Journal of Combinatorics ◽

10.19184/ijc.2021.5.2.7 ◽

2021 ◽

Vol 5 (2) ◽

pp. 110

Author(s):

Zein Rasyid Himami ◽

Denny Riama Silaban

Keyword(s):

Chromatic Number ◽

Connected Graph ◽

Vertex Coloring ◽

Antimagic Labeling ◽

Minimum Number ◽

Fan Graph ◽

Corona Product

Let G=(V,E) be connected graph. A bijection f : E → {1,2,3,..., |E|} is a local antimagic of G if any adjacent vertices u,v ∈ V satisfies w(u)≠ w(v), where w(u)=∑e∈E(u) f(e), E(u) is the set of edges incident to u. When vertex u is assigned the color w(u), we called it a local antimagic vertex coloring of G. A local antimagic chromatic number of G, denoted by χla(G), is the minimum number of colors taken over all colorings induced by the local antimagic labeling of G. In this paper, we determine the local antimagic chromatic number of corona product of friendship and fan with null graph on m vertices, namely, χla(Fn ⊙ \overline{K_m}) and χla(f(1,n) ⊙ \overline{K_m}).

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The local antimagic on disjoint union of some family graphs

Jurnal Matematika MANTIK ◽

10.15642/mantik.2019.5.2.69-75 ◽

2019 ◽

Vol 5 (2) ◽

pp. 69-75

Author(s):

Marsidi Marsidi ◽

Ika Hesti Agustin

Keyword(s):

Lower Bound ◽

Natural Number ◽

Chromatic Number ◽

Disjoint Union ◽

Undirected Graph ◽

Chromatic Numbers ◽

Minimum Number ◽

Vertex Set ◽

Proper Vertex ◽

Vertex Colouring

A graph in this paper is nontrivial, finite, connected, simple, and undirected. Graph consists of a vertex set and edge set. Let u,v be two elements in vertex set, and q is the cardinality of edge set in G, a bijective function from the edge set to the first q natural number is called a vertex local antimagic edge labelling if for any two adjacent vertices and , the weight of is not equal with the weight of , where the weight of (denoted by ) is the sum of labels of edges that are incident to . Furthermore, any vertex local antimagic edge labelling induces a proper vertex colouring on where is the colour on the vertex . The vertex local antimagic chromatic number is the minimum number of colours taken over all colourings induced by vertex local antimagic edge labelling of . In this paper, we discuss about the vertex local antimagic chromatic number on disjoint union of some family graphs, namely path, cycle, star, and friendship, and also determine the lower bound of vertex local antimagic chromatic number of disjoint union graphs. The chromatic numbers of disjoint union graph in this paper attend the lower bound.

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Star chromatic number of Harary graphs

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830921500695 ◽

2020 ◽

pp. 2150069

Author(s):

Chitra Suseendran ◽

Fathima Tabrez

Keyword(s):

Chromatic Number ◽

Vertex Coloring ◽

Minimum Number ◽

Star Coloring ◽

Proper Vertex

A proper vertex coloring of a graph [Formula: see text] is called a star coloring if every path on four vertices in [Formula: see text] is not 2-colored. The star chromatic number is the minimum number of colors required to star color [Formula: see text] and it is denoted by [Formula: see text]. In this paper, the star coloring of Harary graphs [Formula: see text], where [Formula: see text] is even and [Formula: see text] is odd, is discussed.

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Distinguishability of Locally Finite Trees

The Electronic Journal of Combinatorics ◽

10.37236/947 ◽

2007 ◽

Vol 14 (1) ◽

Cited By ~ 11

Author(s):

Mark E. Watkins ◽

Xiangqian Zhou

Keyword(s):

Positive Integer ◽

Chromatic Number ◽

Inverse Image ◽

Vertex Coloring ◽

Finite Tree ◽

Locally Finite ◽

Distinguishing Number ◽

Nonidentity Element ◽

Locally Countable ◽

Proper Vertex

The distinguishing number $\Delta(X)$ of a graph $X$ is the least positive integer $n$ for which there exists a function $f:V(X)\to\{0,1,2,\cdots,n-1\}$ such that no nonidentity element of $\hbox{Aut}(X)$ fixes (setwise) every inverse image $f^{-1}(k)$, $k\in\{0,1,2,\cdots,n-1\}$. All infinite, locally finite trees without pendant vertices are shown to be 2-distinguishable. A proof is indicated that extends 2-distinguishability to locally countable trees without pendant vertices. It is shown that every infinite, locally finite tree $T$ with finite distinguishing number contains a finite subtree $J$ such that $\Delta(J)=\Delta(T)$. Analogous results are obtained for the distinguishing chromatic number, namely the least positive integer $n$ such that the function $f$ is also a proper vertex-coloring.

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On the b-chromatic number of some graph products

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/sscmath.49.2012.2.1194 ◽

2012 ◽

Vol 49 (2) ◽

pp. 156-169 ◽

Cited By ~ 4

Author(s):

Marko Jakovac ◽

Iztok Peterin

Keyword(s):

Chromatic Number ◽

Upper And Lower Bounds ◽

Vertex Coloring ◽

Graph Products ◽

Lexicographic Product ◽

Arbitrary Graph ◽

Strong Product ◽

Color Class ◽

Complete Bipartite Graphs ◽

Proper Vertex

A b-coloring is a proper vertex coloring of a graph such that each color class contains a vertex that has a neighbor in all other color classes and the b-chromatic number is the largest integer φ(G) for which a graph has a b-coloring with φ(G) colors. We determine some upper and lower bounds for the b-chromatic number of the strong product G ⊠ H, the lexicographic product G[H] and the direct product G × H and give some exact values for products of paths, cycles, stars, and complete bipartite graphs. We also show that the b-chromatic number of Pn ⊠ H, Cn ⊠ H, Pn[H], Cn[H], and Km,n[H] can be determined for an arbitrary graph H, when integers m and n are large enough.

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A Study on Harmonious Coloring of Circulant Networks

International Journal of Engineering & Technology ◽

10.14419/ijet.v7i4.10.20945 ◽

2018 ◽

Vol 7 (4.10) ◽

pp. 393

Author(s):

Franklin Thamil Selvi.M.S ◽

Amutha A ◽

Antony Mary A

Keyword(s):

Chromatic Number ◽

Simple Graph ◽

Vertex Coloring ◽

Circulant Networks ◽

Proper Vertex

Given a simple graph , a harmonious coloring of is the proper vertex coloring such that each pair of colors seems to appears together on at most one edge. The harmonious chromatic number of , denoted by is the minimal number of colors in a harmonious coloring of . In this paper we have determined the harmonious chromatic number of some classes of Circulant Networks.

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Star chromatic number of some graphs

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830921500890 ◽

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.

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