scholarly journals On the total vertex irregularity strength of comb product of two cycles and two stars

2020 ◽  
Vol 3 (2) ◽  
pp. 79
Author(s):  
Rismawati Ramdani
Keyword(s):  
Positive Integer ◽  
Connected Graphs ◽  
Vertex Weight ◽  
Irregularity Strength

Let <em>G </em>= (<em>V</em>(<em>G</em>),<em>E</em>(<em>G</em>)) be a graph and <em>k</em> be a positive integer. A total <em>k</em>-labeling of <em>G</em> is a map <em>f</em> : <em>V</em> ∪ <em>E</em> → {1,2,3,...,<em>k</em>}. The vertex weight <em>v</em> under the labeling <em>f</em> is denoted by w_<em>f</em>(<em>v</em>) and defined by <em>w</em>_<em>f</em>(<em>v</em>) = <em>f</em>(<em>v</em>) + \sum_{uv \in{E(G)}} {<em>f</em>(<em>uv</em>)}. A total <em>k</em>-labeling of <em>G</em> is called vertex irregular if there are no two vertices with the same weight. The total vertex irregularity strength of <em>G</em>, denoted by <em>tvs</em>(<em>G</em>), is the minimum <em>k</em> such that <em>G</em> has a vertex irregular total <em>k</em>-labeling. This labelings were introduced by Baca, Jendrol, Miller, and Ryan in 2007. Let <em>G</em> and <em>H</em> be two connected graphs. Let <em>o</em> be a vertex of <em>H</em>. The comb product between <em>G</em> and <em>H</em>, denoted by <em>G </em>\rhd_o <em>H</em>, is a graph obtained by taking one copy of <em>G</em> and |<em>V</em>(<em>G</em>)| copies of <em>H</em> and grafting the i-th copy of <em>H</em> at the vertex <em>o</em> to the i-th vertex of <em>G</em>. In this paper, we determine the total vertex irregularity strength of comb product of two cycles and two stars.

scholarly journals Total vertex irregularity strength of comb product of two cycles

2018 ◽  
Vol 197 ◽  
pp. 01007
Author(s):  
Rismawati Ramdani ◽  
Muhammad Ali Ramdhani
Keyword(s):  
Positive Integer ◽  
Connected Graphs ◽  
Vertex Weight ◽  
Irregularity Strength

Let G = (V (G),E(G)) be a graph and k be a positive integer. A total k-labeling of G is a map f : V (G) ∪ E(G) → {1,2...,k}. The vertex weight v under the labeling f is denoted by Wf(v) and defined by Wf(v) = f(v) + Σuv∈E(G)f(uv). A total k-labeling of G is called vertex irregular if there are no two vertices with the same weight. The total vertex irregularity strength of G, denoted by tvs(G), is the minimum k such that G has a vertex irregular total k-labeling. This labeling was introduced by Bača, Jendrol', Miller, and Ryan in 2007. Let G and H be two connected graphs. Let o be a vertex of H . The comb product between G and H, in the vertex o, denoted by G⊳o H, is a graph obtained by taking one copy of G and |V (G)| copies of H and grafting the i-th copy of H at the vertex o to the i-th vertex of G. In this paper, we determine the total vertex irregularity strength of comb product of Cn and Cm where m ∈ {1,2}.

scholarly journals On The Total Edge and Vertex Irregularity Strength of Some Graphs Obtained from Star

2019 ◽  
Vol 25 (3) ◽  
pp. 314-324
Author(s):  
Rismawati Ramdani ◽  
A.N.M Salman ◽  
Hilda Assiyatun
Keyword(s):  
Positive Integer ◽  
Edge Weight ◽  
Vertex Weight ◽  
Irregularity Strength ◽  
Total Edge Irregularity Strength

Let $G=(V(G),E(G))$ be a graph and $k$ be a positive integer. A total $k$-labeling of $G$ is a map $f: V(G)\cup E(G)\rightarrow \{1,2,\ldots,k \}$. The edge weight $uv$ under the labeling $f$ is denoted by $w_f(uv)$ and defined by $w_f(uv)=f(u)+f(uv)+f(v)$. The vertex weight $v$ under the labeling $f$ is denoted by $w_f(v)$ and defined by $w_f(v) = f(v) + \sum_{uv \in{E(G)}} {f(uv)}$. A total $k$-labeling of $G$ is called an edge irregular total $k$-labeling of $G$ if  $w_f(e_1)\neq w_f(e_2)$ for every two distinct edges $e_1$ and $e_2$  in $E(G)$.  The total edge irregularity strength of $G$, denoted by $tes(G)$, is the minimum $k$ for which $G$ has an edge irregular total $k$-labeling.  A total $k$-labeling of $G$ is called a vertex irregular total $k$-labeling of $G$ if  $w_f(v_1)\neq w_f(v_2)$ for every two distinct vertices $v_1$ and $v_2$ in $V(G)$.  The total vertex irregularity strength of $G$, denoted by $tvs(G)$, is the minimum $k$ for which $G$ has a vertex irregular total $k$-labeling.  In this paper, we determine the total edge irregularity strength and the total vertex irregularity strength of some graphs obtained from star, which are gear, fungus, and some copies of stars.

scholarly journals On generalized 3-connectivity of the strong product of graphs

2018 ◽  
Vol 12 (2) ◽  
pp. 297-317
Author(s):  
Encarnación Abajo ◽  
Rocío Casablanca ◽  
Ana Diánez ◽  
Pedro García-Vázquez
Keyword(s):  
Positive Integer ◽  
Connected Graph ◽  
Sharp Bound ◽  
Connected Graphs ◽  
Strong Product ◽  
Generalized Connectivity ◽  
Internally Disjoint Trees ◽  
Product Of Graphs

Let G be a connected graph with n vertices and let k be an integer such that 2 ? k ? n. The generalized connectivity kk(G) of G is the greatest positive integer l for which G contains at least l internally disjoint trees connecting S for any set S ? V (G) of k vertices. We focus on the generalized connectivity of the strong product G1 _ G2 of connected graphs G1 and G2 with at least three vertices and girth at least five, and we prove the sharp bound k3(G1 _ G2) ? k3(G1)_3(G2) + k3(G1) + k3(G2)-1.

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 .

Some results about component factors in graphs

2019 ◽  
Vol 53 (3) ◽  
pp. 723-730 ◽  
Author(s):  
Sizhong Zhou
Keyword(s):  
Positive Integer ◽  
Connected Graph ◽  
Connected Graphs ◽  
Spanning Subgraph ◽  
Component Factor ◽  
Path Factor

For a set ℋ of connected graphs, a spanning subgraph H of a graph G is called an ℋ-factor of G if every component of H is isomorphic to a member ofℋ. An H-factor is also referred as a component factor. If each component of H is a star (resp. path), H is called a star (resp. path) factor. By a P≥ k-factor (k positive integer) we mean a path factor in which each component path has at least k vertices (i.e. it has length at least k − 1). A graph G is called a P≥ k-factor covered graph, if for each edge e of G, there is a P≥ k-factor covering e. In this paper, we prove that (1) a graph G has a {K1,1,K1,2, … ,K1,k}-factor if and only if bind(G) ≥ 1/k, where k ≥ 2 is an integer; (2) a connected graph G is a P≥ 2-factor covered graph if bind(G) > 2/3; (3) a connected graph G is a P≥ 3-factor covered graph if bind(G) ≥ 3/2. Furthermore, it is shown that the results in this paper are best possible in some sense.

scholarly journals Group irregularity strength of connected graphs

2013 ◽  
Vol 30 (1) ◽  
pp. 1-17 ◽  
Author(s):  
Marcin Anholcer ◽  
Sylwia Cichacz ◽  
Martin Milanic̆
Keyword(s):  
Connected Graphs ◽  
Irregularity Strength

scholarly journals Leafy spanning k-forests

2021 ◽  
Author(s):  
Cristina G. Fernandes ◽  
Carla N. Lintzmayer ◽  
Mário César San Felice
Keyword(s):  
Positive Integer ◽  
Approximation Algorithm ◽  
Best Approximation ◽  
Np Hard ◽  
Connected Graphs ◽  
Spanning Forest ◽  
The Best Approximation ◽  
Number Of Leaves

We denote by Maximum Leaf Spanning k-Forest the problem of, given a positive integer k and a graph G with at most k components, finding a spanning forest in G with at most k components and the maximum number of leaves. A leaf in a forest is defined as a vertex of degree at most one. The case k = 1 for connected graphs is known to be NP-hard, and is well studied in the literature, with the best approximation algorithm proposed more than 20 years ago by Solis-Oba. The best known approximation algorithm for Maximum Leaf Spanning k-Forest with a slightly different leaf definition is a 3-approximation based on an approach by Lu and Ravi for the k = 1 case. We extend the algorithm of Solis-Oba to achieve a 2-approximation for Maximum Leaf Spanning k-Forest.

scholarly journals On the Total Vertex Irregular Labeling of Proper Interval Graphs

2020 ◽  
Vol 12 (4) ◽  
pp. 537-543
Author(s):  
A. Rana
Keyword(s):  
Efficient Algorithm ◽  
Simple Graph ◽  
Interval Graphs ◽  
Vertex Weight ◽  
Vertex Set ◽  
Positive Integers ◽  
Irregularity Strength

A labeling of a graph is a mapping that maps some set of graph elements to a set of numbers (usually positive integers).  For a simple graph G = (V, E) with vertex set V and edge set E, a labeling  Φ: V ∪ E → {1, 2, ..., k} is called total k-labeling. The associated vertex weight of a vertex x∈ V under a total k-labeling  Φ is defined as wt(x) = Φ(x) + ∑y∈N(x) Φ(xy) where N(x) is the set of neighbors of the vertex x. A total k-labeling is defined to be a vertex irregular total labeling of a graph, if for every two different vertices x and y of G, wt(x)≠wt(y). The minimum k for which  a graph G has a vertex irregular total k-labeling is called the total vertex irregularity strength of G, tvs(G). In this paper, total vertex irregularity strength of interval graphs is studied. In particular, an efficient algorithm is designed to compute tvs of proper interval graphs and bounds of tvs is presented for interval graphs.

Herscovici’s conjecture on C2n × G

2020 ◽  
Vol 12 (05) ◽  
pp. 2050071
Author(s):  
A. Lourdusamy ◽  
T. Mathivanan
Keyword(s):  
Positive Integer ◽  
Connected Graph ◽  
Adjacent Vertex ◽  
Connected Graphs ◽  
Number Formula ◽  
Pebbling Number

The [Formula: see text]-pebbling number, [Formula: see text], of a connected graph [Formula: see text], is the smallest positive integer such that from every placement of [Formula: see text] pebbles, [Formula: see text] pebbles can be moved to any specified target vertex by a sequence of pebbling moves, each move taking two pebbles off a vertex and placing one on an adjacent vertex. A graph [Formula: see text] satisfies the [Formula: see text]-pebbling property if [Formula: see text] pebbles can be moved to any specified vertex when the total starting number of pebbles is [Formula: see text], where [Formula: see text] is the number of vertices with at least one pebble. We show that the cycle [Formula: see text] satisfies the [Formula: see text]-pebbling property. Herscovici conjectured that for any connected graphs [Formula: see text] and [Formula: see text], [Formula: see text]. We prove Herscovici’s conjecture is true, when [Formula: see text] is an even cycle and for variety of graphs [Formula: see text] which satisfy the [Formula: see text]-pebbling property.

scholarly journals Major n-connected graphs

1989 ◽  
Vol 47 (1) ◽  
pp. 43-52
Author(s):  
Ortrud R. Oellermann
Keyword(s):  
Positive Integer ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Connected Subgraph ◽  
Edge Connectivity ◽  
Maximum Order ◽  
Induced Subgraph ◽  
Connected Graphs

AbstractAn induced subgraph H of connectivity (edge-connectivity) n in a graph G is a major n-connected (major n-edge-connected) subgraph of G if H contains no subgraph with connectivity (edge- connectivity) exceeding n and H has maximum order with respect to this property. An induced subgraph is a major (major edge-) subgraph if it is a major n-connected (major n-edge-connected) subgraph for some n. Let m be the maximum order among all major subgraphs of C. Then the major connectivity set K(G) of G is defined as the set of all n for which there exists a major n-connected subgraph of G having order m. The major edge-connectivity set is defined analogously. The connectivity and the elements of the major connectivity set of a graph are compared, as are the elements of the major connectivity set and the major edge-connectivity set of a graph. It is shown that every set S of nonnegative integers is the major connectivity set of some graph G. Further, it is shown that for each positive integer m exceeding every element of S, there exists a graph G such that every major k-connected subgraph of G, where k ∈ K(G), has order m. Moreover, upper and lower bounds on the order of such graphs G are established.

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