scholarly journals On 1-rotational decompositions of complete graphs into tripartite graphs

2019 ◽  
Vol 39 (5) ◽  
pp. 623-643
Author(s):  
Ryan C. Bunge
Keyword(s):  
Positive Integer ◽  
Simple Graph ◽  
Vertex Coloring ◽  
Complete Graphs ◽  
Tripartite Graph ◽  
Tripartite Graphs ◽  
Proper Vertex

Consider a tripartite graph to be any simple graph that admits a proper vertex coloring in at most 3 colors. Let \(G\) be a tripartite graph with \(n\) edges, one of which is a pendent edge. This paper introduces a labeling on such a graph \(G\) used to achieve 1-rotational \(G\)-decompositions of \(K_{2nt}\) for any positive integer \(t\). It is also shown that if \(G\) with a pendent edge is the result of adding an edge to a path on \(n\) vertices, then \(G\) admits such a labeling.

scholarly journals Distinguishability of Locally Finite Trees

10.37236/947 ◽  
2007 ◽  
Vol 14 (1) ◽  
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.

scholarly journals A Study on Harmonious Coloring of Circulant Networks

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.  

The Harmonious Chromatic Number of Bounded Degree Trees

1996 ◽  
Vol 5 (1) ◽  
pp. 15-28 ◽  
Author(s):  
Keith Edwards
Keyword(s):  
Positive Integer ◽  
Chromatic Number ◽  
Maximum Degree ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Simple Graph ◽  
Bounded Degree ◽  
Bounded Degree Trees ◽  
Fixed Positive Integer ◽  
Proper Vertex

A harmonious colouring of a simple graph G is a proper vertex colouring such that each pair of colours appears together on at most one edge. The harmonious chromatic number h(G) is the least number of colours in such a colouring. Let d be a fixed positive integer. We show that there is a natural number N(d) such that if T is any tree with m ≥ N(d) edges and maximum degree at most d, then the harmonious chromatic number h(T) is k or k + 1, where k is the least positive integer such that . We also give a polynomial time algorithm for determining the harmonious chromatic number of a tree with maximum degree at most d.

scholarly journals Tiling Tripartite Graphs with $3$-Colorable Graphs

10.37236/198 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Ryan Martin ◽  
Yi Zhao
Keyword(s):  
Positive Integer ◽  
Positive Real Number ◽  
Minimum Degree ◽  
Discrete Math ◽  
The Other ◽  
Sufficient Condition ◽  
Positive Real ◽  
Tripartite Graph ◽  
Colorable Graph ◽  
Tripartite Graphs

For any positive real number $\gamma$ and any positive integer $h$, there is $N_0$ such that the following holds. Let $N\ge N_0$ be such that $N$ is divisible by $h$. If $G$ is a tripartite graph with $N$ vertices in each vertex class such that every vertex is adjacent to at least $(2/3+ \gamma) N$ vertices in each of the other classes, then $G$ can be tiled perfectly by copies of $K_{h,h,h}$. This extends the work in [Discrete Math. 254 (2002), 289–308] and also gives a sufficient condition for tiling by any fixed 3-colorable graph. Furthermore, we show that the minimum-degree $(2/3+ \gamma) N$ in our result cannot be replaced by $2N/3+ h-2$.

scholarly journals The Harmonious Chromatic Number of Almost All Trees

1995 ◽  
Vol 4 (1) ◽  
pp. 31-46 ◽  
Author(s):  
Keith Edwards
Keyword(s):  
Positive Integer ◽  
Chromatic Number ◽  
Simple Graph ◽  
Almost All ◽  
Proper Vertex ◽  
Vertex Colouring

A harmonious colouring of a simple graph G is a proper vertex colouring such that each pair of colours appears together on at most one edge. The harmonious chromatic number h(G) is the least number of colours in such a colouring.For any positive integer m, let Q(m) be the least positive integer k such that ≥ m. We show that for almost all unlabelled, unrooted trees T, h(T) = Q(m), where m is the number of edges of T.

scholarly journals Total Weight Choosability of Graphs

10.37236/599 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Jakub Przybyło ◽  
Mariusz Woźniak
Keyword(s):  
Main Tool ◽  
Bipartite Graphs ◽  
Simple Graph ◽  
Total Weight ◽  
Combinatorial Nullstellensatz ◽  
Complete Graphs ◽  
Complete Bipartite Graphs ◽  
Proper Vertex ◽  
Vertex Colouring

Suppose the edges and the vertices of a simple graph $G$ are assigned $k$-element lists of real weights. By choosing a representative of each list, we specify a vertex colouring, where for each vertex its colour is defined as the sum of the weights of its incident edges and the weight of the vertex itself. How long lists ensures a choice implying a proper vertex colouring for any graph? Is there any finite bound or maybe already lists of length two are sufficient? We prove that $2$-element lists are enough for trees, wheels, unicyclic and complete graphs, while the ones of length $3$ are sufficient for complete bipartite graphs. Our main tool is an algebraic theorem by Alon called Combinatorial Nullstellensatz.

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.

scholarly journals On chromatic Zagreb indices of certain graphs

2017 ◽  
Vol 09 (01) ◽  
pp. 1750014 ◽  
Author(s):  
Johan Kok ◽  
N. K. Sudev ◽  
U. Mary
Keyword(s):  
Connected Graph ◽  
Vertex Coloring ◽  
Zagreb Indices ◽  
Proper Vertex

Let [Formula: see text] be a finite and simple undirected connected graph of order [Formula: see text] and let [Formula: see text] be a proper vertex coloring of [Formula: see text]. Denote [Formula: see text] simply, [Formula: see text]. In this paper, we introduce a variation of the well-known Zagreb indices by utilizing the parameter [Formula: see text] instead of the invariant [Formula: see text] for all vertices of [Formula: see text]. The new indices are called chromatic Zagreb indices. We study these new indices for certain classes of graphs and introduce the notion of chromatically stable graphs.

scholarly journals On Total Vertex Irregularity Strength of Hexagonal Cluster Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Nurdin Hinding ◽  
Hye Kyung Kim ◽  
Nurtiti Sunusi ◽  
Riskawati Mise
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Simple Graph ◽  
Vertex Set ◽  
Cluster Graph ◽  
Cluster Graphs ◽  
Irregularity Strength

For a simple graph G with a vertex set V G and an edge set E G , a labeling f : V G ∪ ​ E G ⟶ 1,2 , ⋯ , k is called a vertex irregular total k − labeling of G if for any two different vertices x and y in V G we have w t x ≠ w t y where w t x = f x + ∑ u ∈ V G f x u . The smallest positive integer k such that G has a vertex irregular total k − labeling is called the total vertex irregularity strength of G , denoted by tvs G . The lower bound of tvs G for any graph G have been found by Baca et. al. In this paper, we determined the exact value of the total vertex irregularity strength of the hexagonal cluster graph on n cluster for n ≥ 2 . Moreover, we show that the total vertex irregularity strength of the hexagonal cluster graph on n cluster is 3 n 2 + 1 / 2 .

Sign in / Sign up

Export Citation Format

Share Document