Graham’s pebbling conjecture holds for the product of a graph and a sufficiently large complete bipartite graph

2019 ◽  
Vol 11 (06) ◽  
pp. 1950068
Author(s):  
Nopparat Pleanmani
Keyword(s):  
Bipartite Graph ◽  
Network Optimization ◽  
Connected Graph ◽  
Complete Bipartite Graph ◽  
Fixed Number ◽  
Arbitrary Vertex ◽  
Graph Pebbling ◽  
Vertex Set ◽  
Graham’S Conjecture ◽  
Pebbling Number

A graph pebbling is a network optimization model for the transmission of consumable resources. A pebbling move on a connected graph [Formula: see text] is the process of removing two pebbles from a vertex and placing one of them on an adjacent vertex after configuration of a fixed number of pebbles on the vertex set of [Formula: see text]. The pebbling number of [Formula: see text], denoted by [Formula: see text], is defined to be the least number of pebbles to guarantee that for any configuration of pebbles on [Formula: see text] and arbitrary vertex [Formula: see text], there is a sequence of pebbling movement that places at least one pebble on [Formula: see text]. For connected graphs [Formula: see text] and [Formula: see text], Graham’s conjecture asserted that [Formula: see text]. In this paper, we show that such conjecture holds when [Formula: see text] is a complete bipartite graph with sufficiently large order in terms of [Formula: see text] and the order of [Formula: see text].

When the annihilator-ideal graph is planar or complete bipartite graph

2019 ◽  
Vol 12 (02) ◽  
pp. 1950024
Author(s):  
M. J. Nikmehr ◽  
S. M. Hosseini
Keyword(s):  
Commutative Ring ◽  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Simple Graph ◽  
Annihilator Ideal ◽  
Vertex Set

Let [Formula: see text] be a commutative ring with identity and [Formula: see text] be the set of ideals of [Formula: see text] with nonzero annihilator. The annihilator-ideal graph of [Formula: see text], denoted by [Formula: see text], is a simple graph with the vertex set [Formula: see text], and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. In this paper, we present some results on the bipartite, complete bipartite, outer planar and unicyclic of the annihilator-ideal graphs of a commutative ring. Among other results, bipartite annihilator-ideal graphs of rings are characterized. Also, we investigate planarity of the annihilator-ideal graph and classify rings whose annihilator-ideal graph is planar.

scholarly journals The K -Size Edge Metric Dimension of Graphs

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Tanveer Iqbal ◽  
Muhammad Naeem Azhar ◽  
Syed Ahtsham Ul Haq Bokhary
Keyword(s):  
Bipartite Graph ◽  
Connected Graph ◽  
Complete Bipartite Graph ◽  
Petersen Graph ◽  
Metric Dimension ◽  
Resolving Set ◽  
Generalized Petersen Graph

In this paper, a new concept k -size edge resolving set for a connected graph G in the context of resolvability of graphs is defined. Some properties and realizable results on k -size edge resolvability of graphs are studied. The existence of this new parameter in different graphs is investigated, and the k -size edge metric dimension of path, cycle, and complete bipartite graph is computed. It is shown that these families have unbounded k -size edge metric dimension. Furthermore, the k-size edge metric dimension of the graphs Pm □ Pn, Pm □ Cn for m, n ≥ 3 and the generalized Petersen graph is determined. It is shown that these families of graphs have constant k -size edge metric dimension.

Some Results on Annihilating-ideal Graphs

2016 ◽  
Vol 59 (3) ◽  
pp. 641-651
Author(s):  
Farzad Shaveisi
Keyword(s):  
Direct Summand ◽  
Commutative Ring ◽  
Bipartite Graph ◽  
Maximum Degree ◽  
Complete Bipartite Graph ◽  
Independence Number ◽  
Prime Ideals ◽  
Reduced Ring ◽  
Vertex Set ◽  
Minimal Prime

AbstractThe annihilating-ideal graph of a commutative ring R, denoted by 𝔸𝔾(R), is a graph whose vertex set consists of all non-zero annihilating ideals and two distinct vertices I and J are adjacent if and only if IJ = (0). Here we show that if R is a reduced ring and the independence number of 𝔸𝔾(R) is finite, then the edge chromatic number of 𝔸𝔾(R) equals its maximum degree and this number equals 2|Min(R)|−1 also, it is proved that the independence number of 𝔸𝔾(R) equals 2|Min(R)|−1, where Min(R) denotes the set of minimal prime ideals of R. Then we give some criteria for a graph to be isomorphic with an annihilating-ideal graph of a ring. For example, it is shown that every bipartite annihilating-ideal graph is a complete bipartite graph with at most two horns. Among other results, it is shown that a ûnite graph 𝔸𝔾(R) is not Eulerian, and that it is Hamiltonian if and only if R contains no Gorenstain ring as its direct summand.

scholarly journals The 2-pebbling property of squares of paths and Graham’s conjecture

2020 ◽  
Vol 18 (1) ◽  
pp. 87-92
Author(s):  
Yueqing Li ◽  
Yongsheng Ye
Keyword(s):  
Connected Graph ◽  
Adjacent Vertex ◽  
Graham’S Conjecture ◽  
Pebbling Number

Abstract A pebbling move on a graph G consists of taking two pebbles off one vertex and placing one pebble on an adjacent vertex. The pebbling number of a connected graph G, denoted by f(G), is the least n such that any distribution of n pebbles on G allows one pebble to be moved to any specified vertex by a sequence of pebbling moves. In this paper, we determine the 2-pebbling property of squares of paths and Graham’s conjecture on $\begin{array}{} P_{2n}^2 \end{array} $.

scholarly journals Mallows permutations as stable matchings

2020 ◽  
pp. 1-25
Author(s):  
Omer Angel ◽  
Alexander E. Holroyd ◽  
Tom Hutchcroft ◽  
Avi Levy
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Stable Matching ◽  
Countable Family ◽  
Stable Matchings ◽  
Uncountable Family ◽  
Edge Crossing ◽  
Vertex Set ◽  
Infinite Intervals ◽  
The Law

Abstract We show that the Mallows measure on permutations of $1,\dots ,n$ arises as the law of the unique Gale–Shapley stable matching of the random bipartite graph with vertex set conditioned to be perfect, where preferences arise from the natural total ordering of the vertices of each gender but are restricted to the (random) edges of the graph. We extend this correspondence to infinite intervals, for which the situation is more intricate. We prove that almost surely, every stable matching of the random bipartite graph obtained by performing Bernoulli percolation on the complete bipartite graph $K_{{\mathbb Z},{\mathbb Z}}$ falls into one of two classes: a countable family $(\sigma _n)_{n\in {\mathbb Z}}$ of tame stable matchings, in which the length of the longest edge crossing k is $O(\log |k|)$ as $k\to \pm \infty $ , and an uncountable family of wild stable matchings, in which this length is $\exp \Omega (k)$ as $k\to +\infty $ . The tame stable matching $\sigma _n$ has the law of the Mallows permutation of ${\mathbb Z}$ (as constructed by Gnedin and Olshanski) composed with the shift $k\mapsto k+n$ . The permutation $\sigma _{n+1}$ dominates $\sigma _{n}$ pointwise, and the two permutations are related by a shift along a random strictly increasing sequence.

scholarly journals The IC-Indices of Complete Bipartite Graphs

10.37236/767 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Chin-Lin Shiue ◽  
Hung-Lin Fu
Keyword(s):  
Bipartite Graph ◽  
Connected Graph ◽  
Complete Bipartite Graph ◽  
Bipartite Graphs ◽  
Connected Subgraph ◽  
Maximum Value ◽  
Complete Bipartite Graphs ◽  
Function Mapping

Let $G$ be a connected graph, and let $f$ be a function mapping $V(G)$ into ${\Bbb N}$. We define $f(H)=\sum_{v\in{V(H)}}f(v)$ for each subgraph $H$ of $G$. The function $f$ is called an IC-coloring of $G$ if for each integer $k$ in the set $\{1,2,\cdots,f(G)\}$ there exists an (induced) connected subgraph $H$ of $G$ such that $f(H)=k$, and the IC-index of $G$, $M(G)$, is the maximum value of $f(G)$ where $f$ is an IC-coloring of $G$. In this paper, we show that $M(K_{m,n})=3\cdot2^{m+n-2}-2^{m-2}+2$ for each complete bipartite graph $K_{m,n},\,2\leq m\leq n$.

A note on pebbling bound for a product of Class 0 graphs

2021 ◽  
pp. 2250031
Author(s):  
Nopparat Pleanmani ◽  
Somnuek Worawiset
Keyword(s):  
Upper Bound ◽  
Connected Graph ◽  
Adjacent Vertex ◽  
Arbitrary Vertex ◽  
Pebbling Number

Let [Formula: see text] be a connected graph. For a configuration of pebbles on the vertices of [Formula: see text], a pebbling move on [Formula: see text] is the process of taking two pebbles from a vertex and adding one of them on an adjacent vertex. The pebbling number of [Formula: see text], denoted by [Formula: see text], is the least number of pebbles to guarantee that for any configuration of pebbles on [Formula: see text] and arbitrary vertex [Formula: see text], there is a sequence of pebbling movement that places at least one pebble on [Formula: see text]. The graph [Formula: see text] is said to be of Class 0 if its pebbling number equals its order. For a Class [Formula: see text] connected graph [Formula: see text], we improve a recent upper bound for [Formula: see text] in terms of [Formula: see text].

scholarly journals The 2-Pebbling Property of the Middle Graph of Fan Graphs

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Yongsheng Ye ◽  
Fang Liu ◽  
Caixia Shi
Keyword(s):  
Connected Graph ◽  
Adjacent Vertex ◽  
Arbitrary Vertex ◽  
Pebbling Number ◽  
Middle Graph

A pebbling move on a graphGconsists of taking two pebbles off one vertex and placing one pebble on an adjacent vertex. The pebbling number of a connected graphG, denoted byf(G), is the leastnsuch that any distribution ofnpebbles onGallows one pebble to be moved to any specified but arbitrary vertex by a sequence of pebbling moves. This paper determines the pebbling numbers and the 2-pebbling property of the middle graph of fan graphs.

scholarly journals The Smith and Critical Groups of the Square Rook's Graph and its Complement

10.37236/5442 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Joshua E. Ducey ◽  
Jonathan Gerhard ◽  
Noah Watson
Keyword(s):  
Normal Form ◽  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Line Graph ◽  
Cartesian Product ◽  
Laplacian Matrix ◽  
Critical Group ◽  
Critical Groups ◽  
Complete Graphs ◽  
Vertex Set

Let $R_{n}$ denote the graph with vertex set consisting of the squares of an $n \times n$ grid, with two squares of the grid adjacent when they lie in the same row or column.  This is the square rook's graph, and can also be thought of as the Cartesian product of two complete graphs of order $n$, or the line graph of the complete bipartite graph $K_{n,n}$.  In this paper we compute the Smith group and critical group of the graph $R_{n}$ and its complement.  This is equivalent to determining the Smith normal form of both the adjacency and Laplacian matrix of each of these graphs.  In doing so we verify a 1986 conjecture of Rushanan.

scholarly journals Codegree Threshold for Tiling Balanced Complete $3$-Partite $3$-Graphs and Generalized $4$-Cycles

10.37236/9061 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Xinmin Hou ◽  
Boyuan Liu ◽  
Yue Ma
Keyword(s):  
Bipartite Graph ◽  
Error Term ◽  
Complete Bipartite Graph ◽  
Linear Term ◽  
Asymptotic Values ◽  
F Factor ◽  
Vertex Set ◽  
Error Terms ◽  
Vertex Disjoint

Given two $k$-graphs $F$ and $H$, a perfect $F$-tiling (also called an $F$-factor) in $H$ is a set of vertex-disjoint copies of $F$ that together cover the vertex set of $H$. Let $t_{k-1}(n, F)$ be the smallest integer $t$ such that every  $k$-graph $H$ on $n$ vertices with minimum codegree at least $t$ contains a perfect $F$-tiling.  Mycroft (JCTA, 2016) determined  the asymptotic values of $t_{k-1}(n, F)$ for $k$-partite $k$-graphs $F$ and conjectured that the error terms $o(n)$ in $t_{k-1}(n, F)$ can be replaced by a constant that depends only on $F$. In this paper, we determine the exact value of $t_2(n, K_{m,m}^{3})$, where $K_{m,m}^{3}$ (defined by Mubayi and Verstraëte, JCTA, 2004) is the 3-graph obtained from the complete bipartite graph $K_{m,m}$ by replacing each vertex in one part by a 2-elements set. Note that $K_{2,2}^{3}$ is  the well known  generalized 4-cycle $C_4^3$ (the 3-graph on six vertices and four distinct edges $A, B, C, D$ with $A\cup B= C\cup D$ and $A\cap B=C\cap D=\emptyset$). The result confirms Mycroft's conjecture for $K_{m,m}^{3}$. Moreover, we improve the error term $o(n)$ to a sub-linear term when $F=K^3(m)$ and show that the sub-linear term is tight for $K^3(2)$, where $K^3(m)$ is the complete $3$-partite $3$-graph with each part of size $m$.

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