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 $m$.

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.

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 Codegree Conditions for Tiling Complete k-Partite k-Graphs and Loose Cycles

2019 ◽  
Vol 28 (06) ◽  
pp. 840-870 ◽  
Author(s):  
Wei Gao ◽  
Jie Han ◽  
Yi Zhao
Keyword(s):  
Error Term ◽  
Uniform Hypergraphs ◽  
Turán Number ◽  
Vertex Set ◽  
Vertex Disjoint

AbstractGiven two k-graphs (k-uniform hypergraphs) F and H, a perfect F-tiling (or F-factor) in H is a set of vertex-disjoint copies of F that together cover the vertex set of H. For all complete k-partite k-graphs K, Mycroft proved a minimum codegree condition that guarantees a K-factor in an n-vertex k-graph, which is tight up to an error term o(n). In this paper we improve the error term in Mycroft’s result to a sublinear term that relates to the Turán number of K when the differences of the sizes of the vertex classes of K are co-prime. Furthermore, we find a construction which shows that our improved codegree condition is asymptotically tight in infinitely many cases, thus disproving a conjecture of Mycroft. Finally, we determine exact minimum codegree conditions for tiling K(k)(1, … , 1, 2) and tiling loose cycles, thus generalizing the results of Czygrinow, DeBiasio and Nagle, and of Czygrinow, respectively.

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 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.

The coloring of the cozero-divisor graph of a commutative ring

2020 ◽  
Vol 12 (03) ◽  
pp. 2050023
Author(s):  
S. Akbari ◽  
S. Khojasteh
Keyword(s):  
Commutative Ring ◽  
Bipartite Graph ◽  
Local Ring ◽  
Chromatic Number ◽  
Upper Bound ◽  
Complete Bipartite Graph ◽  
Clique Number ◽  
Vertex Set ◽  
Finite Commutative Ring ◽  
Non Local

Let [Formula: see text] be a commutative ring with unity. The cozero-divisor graph of [Formula: see text] denoted by [Formula: see text] is a graph with the vertex set [Formula: see text], where [Formula: see text] is the set of all nonzero and non-unit elements of [Formula: see text], and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] and [Formula: see text]. Let [Formula: see text] and [Formula: see text] denote the clique number and the chromatic number of [Formula: see text], respectively. In this paper, we prove that if [Formula: see text] is a finite commutative ring, then [Formula: see text] is perfect. Also, we prove that if [Formula: see text] is a commutative Artinian non-local ring and [Formula: see text] is finite, then [Formula: see text]. For Artinian local ring, we obtain an upper bound for the chromatic number of cozero-divisor graph. Among other results, we prove that if [Formula: see text] is a commutative ring, then [Formula: see text] is a complete bipartite graph if and only if [Formula: see text], where [Formula: see text] and [Formula: see text] are fields. Moreover, we present some results on the complete [Formula: see text]-partite cozero-divisor graphs.

Hamiltonian Cycles in Products of Graphs

1975 ◽  
Vol 17 (5) ◽  
pp. 763-765 ◽  
Author(s):  
Joseph Zaks
Keyword(s):  
Bipartite Graph ◽  
Complete Graph ◽  
Hamiltonian Cycle ◽  
Complete Bipartite Graph ◽  
Hamiltonian Path ◽  
Hamiltonian Cycles ◽  
Vertex Set ◽  
Cycle Path

Let V(G) and E(G) denote the vertex set and the edge set of a graph G; let Kn denote the complete graph with n vertices and let Kn, m denote the complete bipartite graph on n and m vertices. A Hamiltonian cycle (Hamiltonian path, respectively) in a graph G is a cycle (path, respectively) in G that contains all the vertices of G.

scholarly journals Partitioning the vertex set of $G$ to make $G\,\Box\, H$ an efficient open domination graph

2016 ◽  
Vol Vol. 18 no. 3 (Graph Theory) ◽  
Author(s):  
Tadeja Kraner Šumenjak ◽  
Iztok Peterin ◽  
Douglas F. Rall ◽  
Aleksandra Tepeh
Keyword(s):  
Bipartite Graph ◽  
Complete Graph ◽  
Complete Bipartite Graph ◽  
Cartesian Product ◽  
Cartesian Products ◽  
Vertex Set

A graph is an efficient open domination graph if there exists a subset of vertices whose open neighborhoods partition its vertex set. We characterize those graphs $G$ for which the Cartesian product $G \Box H$ is an efficient open domination graph when $H$ is a complete graph of order at least 3 or a complete bipartite graph. The characterization is based on the existence of a certain type of weak partition of $V(G)$. For the class of trees when $H$ is complete of order at least 3, the characterization is constructive. In addition, a special type of efficient open domination graph is characterized among Cartesian products $G \Box H$ when $H$ is a 5-cycle or a 4-cycle. Comment: 16 pages, 2 figures

scholarly journals Edit Distance and its Computation

10.37236/744 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
József Balogh ◽  
Ryan Martin
Keyword(s):  
Bipartite Graph ◽  
Edit Distance ◽  
Complete Bipartite Graph ◽  
New Method ◽  
Independent Interest ◽  
Regularity Lemma ◽  
Hereditary Property ◽  
Vertex Set ◽  
Hereditary Properties ◽  
Szemeredi's Regularity Lemma

In this paper, we provide a method for determining the asymptotic value of the maximum edit distance from a given hereditary property. This method permits the edit distance to be computed without using Szemerédi's Regularity Lemma directly. Using this new method, we are able to compute the edit distance from hereditary properties for which it was previously unknown. For some graphs $H$, the edit distance from ${\rm Forb}(H)$ is computed, where ${\rm Forb}(H)$ is the class of graphs which contain no induced copy of graph $H$. Those graphs for which we determine the edit distance asymptotically are $H=K_a+E_b$, an $a$-clique with $b$ isolated vertices, and $H=K_{3,3}$, a complete bipartite graph. We also provide a graph, the first such construction, for which the edit distance cannot be determined just by considering partitions of the vertex set into cliques and cocliques. In the process, we develop weighted generalizations of Turán's theorem, which may be of independent interest.

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