Local Chromatic Number Research Articles

Locating-chromatic number of the edge-amalgamation of trees

<div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p><span>The investigation on the locating-chromatic number of a graph was initiated by Chartrand </span><span>et al. </span><span>(2002). This concept is in fact a special case of the partition dimension of a graph. This topic has received much attention. However, the results are still far from satisfaction. We can define the locating-chromatic number of a graph </span><span>G </span><span>as the smallest integer </span><span>k </span><span>such that there exists a </span><span>k</span><span>-partition of the vertex-set of </span><span>G </span><span>such that all vertices have distinct coordinates with respect to this partition. As we know that the metric dimension of a tree is completely solved. However, the locating-chromatic numbers for most of trees are still open. For </span><span><em>i</em> </span><span>= 1</span><span>, </span><span>2</span><span>, . . . , <em>t</em>, </span><span>let </span><em>T</em><span>i </span><span>be a tree with a fixed edge </span><span>e</span><span>o</span><span>i </span><span>called the terminal edge. The edge-amalgamation of all </span><span>T</span><span>i</span><span>s </span><span>denoted by Edge-Amal</span><span>{</span><span>T</span><span>i</span><span>;</span><span>e</span><span>o</span><span>i</span><span>} </span><span>is a tree formed by taking all the </span><span>T</span><span>i</span><span>s and identifying their terminal edges. In this paper, we study the locating-chromatic number of the edge-amalgamation of arbitrary trees. We give lower and upper bounds for their locating-chromatic numbers and show that the bounds are tight.</span></p></div></div></div>

Improved algorithm for the locating-chromatic number of trees

The concept of the locating-chromatic number for graphs was introduced by Chartrand et al. (2002). In this paper, we propose an algorithm to determine the upper bound of the locating-chromatic number of any tree. This algorithm works much better than the one given by Furuya and Matsumoto (2019).

CERTAIN OPERATION OF GENERALIZED PETERSEN GRAPHS HAVING LOCATING-CHROMATIC NUMBER FIVE

The locating-chromatic number of a graph is combined two graph concept, coloring vertices and partition dimension of a graph. The locating-chromatic number, denoted by XLG, is the smallest k such that G has a locating k-coloring. In this paper, we discuss the locating- chromatic number for certain operation of generalized Petersen graphs sP(n,1).

On The Locating-Chromatic Numbers of Subdivisions of Friendship Graph

Let c be a k-coloring of a connected graph G and let pi={C1,C2,...,Ck} be the partition of V(G) induced by c. For every vertex v of G, let c_pi(v) be the coordinate of v relative to pi, that is c_pi(v)=(d(v,C1 ),d(v,C2 ),...,d(v,Ck )), where d(v,Ci )=min{d(v,x)|x in Ci }. If every two vertices of G have different coordinates relative to pi, then c is said to be a locating k-coloring of G. The locating-chromatic number of G, denoted by chi_L (G), is the least k such that there exists a locating k-coloring of G. In this paper, we determine the locating-chromatic numbers of some subdivisions of the friendship graph Fr_t, that is the graph obtained by joining t copies of 3-cycle with a common vertex, and we give lower bounds to the locating-chromatic numbers of few other subdivisions of Fr_t.

The locating-chromatic number and partition dimension of fibonacene graphs

<p>Fibonacenes are unbranched catacondensed benzenoid hydrocarbons in which all the non-terminal hexagons are angularly annelated. A hexagon is said to be angularly annelated if the hexagon is adjacent to exactly two other hexagons and possesses two adjacent vertices of degree 2. Fibonacenes possess remarkable properties related with Fibonacci numbers. Various graph properties of fibonacenes have been extensively studied, such as their saturation numbers, independence numbers and Wiener indices. In this paper, we show that the locating-chromatic number of any fibonacene graph is 4 and the partition dimension of such a graph is 3.</p>

The Locating-Chromatic Number for Certain Operation of Generalized Petersen Graphs sP(4,2)

The locating-chromatic number of a graph combined two graph concept, coloring vertices and partition dimension of a graph. The locating-chromatic number, denoted by χL(G), is the smallest k such that G has a locating k-coloring. In this paper, we discuss the locating-chromatic number for certain operation of generalized Petersen graphs sP(4,2).

The Locating Chromatic Number of a Disjoint Union of Some Double Stars

In this paper we discuss the locating chromatic number of a disconnected graph, namely a disjoint union of some double stars. To determine the locating-chromatic number of a graph, first we construct the upper bound of the number and observe the minimum coloring used. Then, to determine the lower bound of the number, we use the trivial lower bound for some leaves which incident to one vertex. Two main theorems about the locating chromatic number of a disjoint union of some double stars. We obtained some original results of the locating chromatic number of disjoint union of some double stars. Moreover, we can generalize the graph such that the locating chromatic number remains the same with the previous one.

The locating chromatic number of disconnected graph with path and cycle graph as its components

A special case of the graph partition dimension is The locating-chromatic number of a graph. In 2002, The concept of locating-chromatic number of a connected graph was introduced by Chartrand et al.. In 2014, definition locating-chromatic number for a disconnected graph is extended by Welyyanti et al. As a continuation, we determine the locating-chromatic number of some disconnected graphs with path and cycle as its components.

Calculating an upper bound of the locating-chromatic number of trees

The locating-chromatic number of a graph G(V,E) is the cardinality of a minimum resolving partition of the vertex set V(G) such that all vertices have distinct coordinates with respect to this partition and every two adjacent vertices not contained in the same partition class. Determining the locating-chromatic number of any tree is a difficult task. In this paper, we propose an algorithm to compute the upper bound on the locating-chromatic number of any tree. To do so, we decompose a tree into caterpillars and then compute the upper bound of the locating-chromatic number of this tree in terms of the ones for these caterpillars.

Locating Chromatic Number of Powers of Paths and Cycles

Let c be a proper k-coloring of a graph G. Let π = { R 1 , R 2 , … , R k } be the partition of V ( G ) induced by c, where R i is the partition class receiving color i. The color code c π ( v ) of a vertex v of G is the ordered k-tuple ( d ( v , R 1 ) , d ( v , R 2 ) , … , d ( v , R k ) ) , where d ( v , R i ) is the minimum distance from v to each other vertex u ∈ R i for 1 ≤ i ≤ k . If all vertices of G have distinct color codes, then c is called a locating k-coloring of G. The locating-chromatic number of G, denoted by χ L ( G ) , is the smallest k such that G admits a locating coloring with k colors. In this paper, we give a characterization of the locating chromatic number of powers of paths. In addition, we find sharp upper and lower bounds for the locating chromatic number of powers of cycles.

Further Results on Locating Chromatic Number for Amalgamation of Stars Linking by One Path

Let G = (V,E) be a connected graph. Let c be a proper coloring using k colors, namely 1, 2,·s, k. Let <span style="font-family: symbol;">P</span>={S<sub>1</sub>, S<sub>2</sub>,..., S<sub>k</sub>} be a partition of V(G) induced by c and let S<sub>i</sub> be the color class that receives the color i. The color code, c<sub><span style="font-family: symbol;">P</span></sub>(v)=(d(v,S<sub>1</sub>), d(v,S<sub>2</sub>),...,d(v,S<sub>k</sub>)), where d(v,S<sub>i</sub>)=min {d(v,x)|x <span style="font-family: symbol;">Î</span> S<sub>i</sub>} for i <span style="font-family: symbol;">Î</span> [1,k]. If all vertices in V(G) have different color codes, then c is called as the \emphlocating-chromatic k-coloring of G. Minimum k such that G has the locating-chromatic k-coloring is called the locating-chromatic number, denoted by <span style="font-family: symbol;">c</span><sub>L</sub>(G). In this paper, we discuss the locating-chromatic number for n certain amalgamation of stars linking a path, denoted by nS<sub>k,m</sub>, for n ≥ 1, m ≥ 2, k ≥ 3, and k>m.

BEBERAPA SYARAT CUKUP UNTUK BILANGAN KROMATIK LOKASI HINGGA PADA GRAF TAK TERHUBUNG

The locating-chromatic number of a graph is introduced by Chartrand et al. in 2002. Firstly, Chatrand et al. determine the locating-chromatic number of path and double stars. The locating-chromatic number is an interesting concept ini graph theory. In this paper, we determine some condtions for disconnected graphs has a finite locating-chromatic number.

The locating-chromatic number of subdivision firecracker graphs

The locating-chromatic number of a graph was combined two graph concept, coloring vertices and partition dimension of a graph. In this paper, we discuss about locating-chromatic number of a subdivision firecracker graphs.

On the locating-chromatic number for graphs with two homogenous components

The locating-chromatic number of a graph was introduced by Chartrand et al. in 2002. The concept of the locating-chromatic number is a marriage between graph coloring and the notion of graph partition dimension. This concept is only for connected graphs. In [8], we extended this concept also for disconnected graphs. In this paper, we determine the locating- chromatic number of a graph with two components. In particular, we determine such values if the components are homogeneous and each component has locating-chromatic number 3.

NP-completeness of local colorings of graphs

A local k-coloring of a graph G is a function f:V(G)→{1,2,⋯,k} such that for each S⊆V(G), 2≤|S|≤3, there exist u,v∈S with |f(u)−f(v)| at least the size of the subgraph induced by S. The local chromatic number of G is χℓ(G)=min⁡{k:G has a local k-coloring}. In this paper, we show that it is NP-complete to determine whether a graph has a local k-coloring for fixed k=4 or k=2t−1, where t≥3. In particular, it is NP-complete to determine whether a planar graph has a local 5-coloring even restricted to the maximum degree Δ=7 or 8.

ON SOME PETERSEN GRAPHS HAVING LOCATING CHROMATIC NUMBER FOUR OR FIVE

In this paper, we determine some Petersen graphs having locating chromatic number four or five. Moreover, we give some conjecture to characterize some Petersen graphs having locating chromatic number four

Colorful Subhypergraphs in Uniform Hypergraphs

There are several topological results ensuring in any properly colored graph the existence of a colorful complete bipartite subgraph, whose order is bounded from below by some topological invariants of some topological spaces associated to the graph. Meunier [Colorful subhypergraphs in Kneser hypergraphs, The Electronic Journal of Combinatorics, 2014] presented the first colorful type result for uniform hypergraphs. In this paper, we give some new generalizations of the $\mathbb{Z}_p$-Tucker lemma and by use of them, we improve Meunier's result and some other colorful results by Simonyi, Tardif, and Zsbán [Colourful theorems and indices of homomorphism complexes, The Electronic Journal of Combinatorics, 2014] and by Simonyi and Tardos [Colorful subgraphs in Kneser-like graphs, European Journal of Combinatorics, 2007] to uniform hypergraphs. Also, we introduce some new lower bounds for the chromatic number and local chromatic number of uniform hypergraphs. A hierarchy between these lower bounds is presented as well.

Local coloring of self complementary graphs

Let G=(V,E) be a graph. A local coloring of a graph G of order at least 2 is a function c:V(G)⟶N having the property that for each set S⊆V(G) with 2≤|S|≤3, there exist vertices u,v∈S such that |c(u)−c(v)|≥ms, where ms is the size of the induced subgraph 〈S〉. The maximum color assigned by a local coloring c to a vertex of G is called the value of c and is denoted by χℓ(c). The local chromatic number of G is χℓ(G)=min{χℓ(c)}, where the minimum is taken over all local colorings c of G. In this paper we study the local coloring for some self complementary graphs. Also we present a sc-graph with local chromatic number k for any given integer k≥6.

Trees with Certain Locating-Chromatic Number

The locating-chromatic number of a graph G can be defined as the cardinality of a minimum resolving partition of the vertex set V(G) such that all vertices have distinct coordinates with respect to this partition and every two adjacent vertices in G are not contained in the same partition class. In this case, the coordinate of a vertex v in G is expressed in terms of the distances of v to all partition classes. This concept is a special case of the graph partition dimension notion. Previous authors have characterized all graphs of order n with locating-chromatic number either n or n-1 . They also proved that there exists a tree of order n , n≥5 , having locating-chromatic number k if and only if k ∈{3,4,…, n -2, n }. In this paper, we characterize all trees of order n with locating-chromatic number n - t , for any integers n and t , where n > t +3 and 2 ≤ t < n /2.

A bound for the locating chromatic number of trees

Let f be a proper k-coloring of a connected graph G and Π = (V1, V2, . . . , Vk) be an ordered partition of V (G) into the resulting color classes. For a vertex v of G, the color code of v with respect to Π is defined to be the ordered k-tuple cΠ(v) = (d(v, V1), d(v, V2), . . . , d(v, Vk)), where d(v, Vi) = min{d(v, x) : x ∈ Vi}, 1 ≤ i ≤ k. If distinct vertices have distinct color codes, then f is called a locating coloring. The minimum number of colors needed in a locating coloring of G is the locating chromatic number of G, denoted by χ L (G). In this paper, we study the locating chromatic numbers of trees. We provide a counter example to a theorem of Gary Chartrand et al. [G. Chartrand, D. Erwin, M.A. Henning, P.J. Slater, P. Zhang, The locating-chromatic number of a graph, Bull. Inst. Combin. Appl. 36 (2002) 89-101] about the locating chromatic number of trees. Also, we offer a new bound for the locating chromatic number of trees. Then, by constructing a special family of trees, we show that this bound is best possible.