scholarly journals On The Locating-Chromatic Numbers of Subdivisions of Friendship Graph

2020 ◽  
Vol 26 (2) ◽  
pp. 175-184
Author(s):  
Brilly Maxel Salindeho ◽  
Hilda Assiyatun ◽  
Edy Tri Baskoro
Keyword(s):  
Chromatic Number ◽  
Lower Bounds ◽  
Connected Graph ◽  
Common Vertex ◽  
Chromatic Numbers

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.

scholarly journals Coloring Chains for Compression with Uncertain Priors

10.37236/6468 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Noah Golowich
Keyword(s):  
Chromatic Number ◽  
Lower Bounds ◽  
Graph Coloring ◽  
Upper And Lower Bounds ◽  
Nice Property ◽  
Chromatic Numbers ◽  
Compression Scheme ◽  
Graph Coloring Problem ◽  
Close Relationship ◽  
Local Graph

Haramaty and Sudan considered the problem of transmitting a message between two people, Alice and Bob, when Alice's and Bob's priors on the message are allowed to differ by at most a given factor. To find a deterministic compression scheme for this problem, they showed that it is sufficient to obtain an upper bound on the chromatic number of a graph, denoted $U(N,s,k)$ for parameters $N,s,k$, whose vertices are nested sequences of subsets and whose edges are between vertices that have similar sequences of sets. In turn, there is a close relationship between the problem of determining the chromatic number of $U(N,s,k)$ and a local graph coloring problem considered by Erdős et al. We generalize the results of Erdős et al. by finding bounds on the chromatic numbers of graphs $H$ and $G$ when there is a homomorphism $\phi :H\rightarrow G$ that satisfies a nice property. We then use these results to improve upper and lower bounds on $\chi(U(N,s,k))$. 

Distance Laplacian eigenvalues and chromatic number in graphs

Filomat ◽  
2017 ◽  
Vol 31 (9) ◽  
pp. 2545-2555 ◽  
Author(s):  
Mustapha Aouchiche ◽  
Pierre Hansen
Keyword(s):  
Spectral Radius ◽  
Chromatic Number ◽  
Lower Bounds ◽  
Connected Graph ◽  
Extremal Graphs ◽  
Laplacian Spectral Radius ◽  
Laplacian Eigenvalues ◽  
Fixed Order

In the present paper we are interested in the study of the distance Laplacian eigenvalues of a connected graph with fixed order n and chromatic number x. We prove lower bounds on the distance Laplacian spectral radius in terms of n and x. We also prove results related to the distribution of the distance Laplacian eigenvalues with respect to the values of the chromatic number x. For some of the results, we characterize the extremal graphs, for others, we give examples of extremal graphs.

scholarly journals Chromatic Numbers of Stable Kneser Hypergraphs via Topological Tverberg-Type Theorems

2018 ◽  
Vol 2020 (13) ◽  
pp. 4037-4061 ◽  
Author(s):  
Florian Frick
Keyword(s):  
Euclidean Space ◽  
Chromatic Number ◽  
Lower Bounds ◽  
Pairwise Disjoint ◽  
Convex Hulls ◽  
Chromatic Numbers ◽  
Set Systems ◽  
Equivariant Topology ◽  
Disjoint Sets ◽  
Combination Of Methods

Abstract Kneser’s 1955 conjecture—proven by Lovász in 1978—asserts that in any partition of the $k$-subsets of $\{1, 2, \dots , n\}$ into $n-2k+1$ parts, one part contains two disjoint sets. Schrijver showed that one can restrict to significantly fewer $k$-sets and still observe the same intersection pattern. Alon, Frankl, and Lovász proved a different generalization of Kneser’s conjecture for $r$ pairwise disjoint sets. Dolnikov generalized Lovász’ result to arbitrary set systems, while Kříž did the same for the $r$-fold extension of Kneser’s conjecture. Here we prove a common generalization of all of these results. Moreover, we prove additional strengthenings by determining the chromatic number of certain sparse stable Kneser hypergraphs, and further develop a general approach to establishing lower bounds for chromatic numbers of hypergraphs using a combination of methods from equivariant topology and intersection results for convex hulls of points in Euclidean space.

scholarly journals On the Generalized Distance Energy of Graphs

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 17 ◽  
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Hilal A. Ganie ◽  
Yilun Shang
Keyword(s):  
Lower Bounds ◽  
Complete Graph ◽  
Connected Graph ◽  
Distance Matrix ◽  
Wiener Index ◽  
Diagonal Matrix ◽  
Upper And Lower Bounds ◽  
Star Graph ◽  
Extremal Graphs ◽  
Generalized Distance

The generalized distance matrix D α ( G ) of a connected graph G is defined as D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where 0 ≤ α ≤ 1 , D ( G ) is the distance matrix and T r ( G ) is the diagonal matrix of the node transmissions. In this paper, we extend the concept of energy to the generalized distance matrix and define the generalized distance energy E D α ( G ) . Some new upper and lower bounds for the generalized distance energy E D α ( G ) of G are established based on parameters including the Wiener index W ( G ) and the transmission degrees. Extremal graphs attaining these bounds are identified. It is found that the complete graph has the minimum generalized distance energy among all connected graphs, while the minimum is attained by the star graph among trees of order n.

Joins of paths and cycles of equal order with sigma chromatic number 2

2021 ◽  
pp. 2250006
Author(s):  
Agnes D. Garciano ◽  
Maria Czarina T. Lagura ◽  
Reginaldo M. Marcelo
Keyword(s):  
Chromatic Number ◽  
Connected Graph ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Odd Cycle ◽  
Necessary And Sufficient ◽  
Positive Integers ◽  
Simple Connected Graph

For a simple connected graph [Formula: see text] let [Formula: see text] be a coloring of [Formula: see text] where two adjacent vertices may be assigned the same color. Let [Formula: see text] be the sum of colors of neighbors of any vertex [Formula: see text] The coloring [Formula: see text] is a sigma coloring of [Formula: see text] if for any two adjacent vertices [Formula: see text] [Formula: see text] The least number of colors required in a sigma coloring of [Formula: see text] is the sigma chromatic number of [Formula: see text] and is denoted by [Formula: see text] A sigma coloring of a graph is a neighbor-distinguishing type of coloring and it is known that the sigma chromatic number of a graph is bounded above by its chromatic number. It is also known that for a path [Formula: see text] and a cycle [Formula: see text] where [Formula: see text] [Formula: see text] and [Formula: see text] if [Formula: see text] is even. Let [Formula: see text] the join of the graphs [Formula: see text], where [Formula: see text] or [Formula: see text] [Formula: see text] and [Formula: see text] is not an odd cycle for any [Formula: see text]. It has been shown that if [Formula: see text] for [Formula: see text] and [Formula: see text] then [Formula: see text]. In this study, we give necessary and sufficient conditions under which [Formula: see text] where [Formula: see text] is the join of copies of [Formula: see text] and/or [Formula: see text] for the same value of [Formula: see text]. Let [Formula: see text] and [Formula: see text] be positive integers with [Formula: see text] and [Formula: see text] In this paper, we show that [Formula: see text] if and only if [Formula: see text] or [Formula: see text] is odd, [Formula: see text] is even and [Formula: see text]; and [Formula: see text] if and only if [Formula: see text] is even and [Formula: see text] We also obtain necessary and sufficient conditions on [Formula: see text] and [Formula: see text], so that [Formula: see text] for [Formula: see text] where [Formula: see text] or [Formula: see text] other than the cases [Formula: see text] and [Formula: see text]

scholarly journals Multiplicative Zagreb indices of cacti

2016 ◽  
Vol 08 (03) ◽  
pp. 1650040 ◽  
Author(s):  
Shaohui Wang ◽  
Bing Wei
Keyword(s):  
Graph Theory ◽  
Lower Bounds ◽  
Connected Graph ◽  
Upper And Lower Bounds ◽  
Extremal Graphs ◽  
Zagreb Index ◽  
Zagreb Indices ◽  
Material Chemistry ◽  
Cactus Graph ◽  
Cactus Graphs

Let [Formula: see text] be multiplicative Zagreb index of a graph [Formula: see text]. A connected graph is a cactus graph if and only if any two of its cycles have at most one vertex in common, which is a generalization of trees and has been the interest of researchers in the field of material chemistry and graph theory. In this paper, we use a new tool to obtain the upper and lower bounds of [Formula: see text] for all cactus graphs and characterize the corresponding extremal graphs.

scholarly journals Distance Domination and Distance Irredundance in Graphs

10.37236/953 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Adriana Hansberg ◽  
Dirk Meierling ◽  
Lutz Volkmann
Keyword(s):  
Lower Bounds ◽  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
Distance Domination

A set $D\subseteq V$ of vertices is said to be a (connected) distance $k$-dominating set of $G$ if the distance between each vertex $u\in V-D$ and $D$ is at most $k$ (and $D$ induces a connected graph in $G$). The minimum cardinality of a (connected) distance $k$-dominating set in $G$ is the (connected) distance $k$-domination number of $G$, denoted by $\gamma_k(G)$ ($\gamma_k^c(G)$, respectively). The set $D$ is defined to be a total $k$-dominating set of $G$ if every vertex in $V$ is within distance $k$ from some vertex of $D$ other than itself. The minimum cardinality among all total $k$-dominating sets of $G$ is called the total $k$-domination number of $G$ and is denoted by $\gamma_k^t(G)$. For $x\in X\subseteq V$, if $N^k[x]-N^k[X-x]\neq\emptyset$, the vertex $x$ is said to be $k$-irredundant in $X$. A set $X$ containing only $k$-irredundant vertices is called $k$-irredundant. The $k$-irredundance number of $G$, denoted by $ir_k(G)$, is the minimum cardinality taken over all maximal $k$-irredundant sets of vertices of $G$. In this paper we establish lower bounds for the distance $k$-irredundance number of graphs and trees. More precisely, we prove that ${5k+1\over 2}ir_k(G)\geq \gamma_k^c(G)+2k$ for each connected graph $G$ and $(2k+1)ir_k(T)\geq\gamma_k^c(T)+2k\geq |V|+2k-kn_1(T)$ for each tree $T=(V,E)$ with $n_1(T)$ leaves. A class of examples shows that the latter bound is sharp. The second inequality generalizes a result of Meierling and Volkmann and Cyman, Lemańska and Raczek regarding $\gamma_k$ and the first generalizes a result of Favaron and Kratsch regarding $ir_1$. Furthermore, we shall show that $\gamma_k^c(G)\leq{3k+1\over2}\gamma_k^t(G)-2k$ for each connected graph $G$, thereby generalizing a result of Favaron and Kratsch regarding $k=1$.

On the Babai and Upper Chromatic Numbers of Graphs of Diameter 2

2021 ◽  
Vol 52 (1) ◽  
pp. 113-123
Author(s):  
Peter Johnson ◽  
Alexis Krumpelman
Keyword(s):  
Chromatic Number ◽  
Metric Space ◽  
Simple Graph ◽  
Chromatic Numbers

The Babai numbers and the upper chromatic number are parameters that can be assigned to any metric space. They can, therefore, be assigned to any connected simple graph. In this paper we make progress in the theory of the first Babai number and the upper chromatic number in the simple graph setting, with emphasis on graphs of diameter 2.

scholarly journals Hyper-Wiener indices of polyphenyl chains and polyphenyl spiders

2019 ◽  
Vol 17 (1) ◽  
pp. 668-676
Author(s):  
Tingzeng Wu ◽  
Huazhong Lü
Keyword(s):  
Lower Bounds ◽  
Connected Graph ◽  
Wiener Index ◽  
Upper And Lower Bounds ◽  
Extremal Graphs ◽  
Recurrence Formulae

Abstract Let G be a connected graph and u and v two vertices of G. The hyper-Wiener index of graph G is $\begin{array}{} WW(G)=\frac{1}{2}\sum\limits_{u,v\in V(G)}(d_{G}(u,v)+d^{2}_{G}(u,v)) \end{array}$, where dG(u, v) is the distance between u and v. In this paper, we first give the recurrence formulae for computing the hyper-Wiener indices of polyphenyl chains and polyphenyl spiders. We then obtain the sharp upper and lower bounds for the hyper-Wiener index among polyphenyl chains and polyphenyl spiders, respectively. Moreover, the corresponding extremal graphs are determined.

A Note on Primitive Graphs

1972 ◽  
Vol 15 (3) ◽  
pp. 437-440 ◽  
Author(s):  
I. Z. Bouwer ◽  
G. F. LeBlanc
Keyword(s):  
Connected Graph ◽  
Common Vertex ◽  
Connected Subgraph ◽  
Property A ◽  
Vertex Set

Let G denote a connected graph with vertex set V(G) and edge set E(G). A subset C of E(G) is called a cutset of G if the graph with vertex set V(G) and edge set E(G)—C is not connected, and C is minimal with respect to this property. A cutset C of G is simple if no two edges of C have a common vertex. The graph G is called primitive if G has no simple cutset but every proper connected subgraph of G with at least one edge has a simple cutset. For any edge e of G, let G—e denote the graph with vertex set V(G) and with edge set E(G)—e.

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