scholarly journals Bipartite graph bundles with connected fibres

1999 ◽  
Vol 59 (1) ◽  
pp. 153-161 ◽  
Author(s):  
Sungpyo Hong ◽  
Jin Ho Kwak ◽  
Jaeun Lee
Keyword(s):  
Bipartite Graph ◽  
Simple Graph ◽  
Isomorphism Classes ◽  
Graph Coverings

Let G be a finite connected simple graph. The isomorphism classes of graph bundles and graph coverings over G have been enumerated by Kwak and Lee. Recently, Archdeacon and others characterised bipartite coverings of G and enumerated the isomorphism classes of regular 2p-fold bipartite coverings of G, when G is nonbipartite. In this paper, we characterise bipartite graph bundles over G and derive some enumeration formulas of the isomorphism classes of them when the fibre is a connected bipartite graph. As an application, we compute the exact numbers of the isomorphism classes of bipartite graph bundles over G when the fibre is the path Pn or the cycle Cn.

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.

Finite commutative rings with higher genus unit graphs

2014 ◽  
Vol 14 (01) ◽  
pp. 1550002 ◽  
Author(s):  
Huadong Su ◽  
Kenta Noguchi ◽  
Yiqiang Zhou
Keyword(s):  
Nonnegative Integer ◽  
Commutative Rings ◽  
Orientable Surface ◽  
Simple Graph ◽  
Isomorphism Classes ◽  
Unit Graph ◽  
Higher Genus ◽  
Vertex Set ◽  
Finite Commutative Rings

Let R be a ring with identity. The unit graph of R, denoted by G(R), is a simple graph with vertex set R, and where two distinct vertices x and y are adjacent if and only if x + y is a unit in R. The genus of a simple graph G is the smallest nonnegative integer g such that G can be embedded into an orientable surface Sg. In this paper, we determine all isomorphism classes of finite commutative rings whose unit graphs have genus at most three.

Isomorphism Classes of Concrete Graph Coverings

1998 ◽  
Vol 11 (2) ◽  
pp. 265-272 ◽  
Author(s):  
Rongquan Feng ◽  
Jin Ho Kwak ◽  
Juyoung Kim ◽  
Jaeun Lee
Keyword(s):  
Isomorphism Classes ◽  
Graph Coverings

INTERSECTION GRAPH OF SUBMODULES OF A MODULE

2012 ◽  
Vol 11 (01) ◽  
pp. 1250019 ◽  
Author(s):  
S. AKBARI ◽  
H. A. TAVALLAEE ◽  
S. KHALASHI GHEZELAHMAD
Keyword(s):  
Bipartite Graph ◽  
Chromatic Number ◽  
Clique Number ◽  
Simple Graph ◽  
Intersection Graph ◽  
Star Graph ◽  
Graph Theoretic ◽  
Nonzero Intersection ◽  
One To One

Let R be a ring with identity and M be a unitary left R-module. The intersection graph of an R-moduleM, denoted by G(M), is defined to be the undirected simple graph whose vertices are in one to one correspondence with all non-trivial submodules of M and two distinct vertices are adjacent if and only if the corresponding submodules of M have nonzero intersection. We investigate the interplay between the module-theoretic properties of M and the graph-theoretic properties of G(M). We characterize all modules for which the intersection graph of submodules is connected. Also the diameter and the girth of G(M) are determined. We study the clique number and the chromatic number of G(M). Among other results, it is shown that if G(M) is a bipartite graph, then G(M) is a star graph.

scholarly journals Reconstructing in the Constraint Satisfaction Problem

10.29007/1j7l ◽  
2020 ◽  
Author(s):  
Evgeny Dantsin
Keyword(s):  
Constraint Satisfaction ◽  
Isomorphism Class ◽  
Existence Of Solutions ◽  
Constraint Satisfaction Problem ◽  
Simple Graph ◽  
Number Of Solutions ◽  
Isomorphism Classes ◽  
Graph Homomorphisms ◽  
Graph Reconstruction ◽  
Standing Problem

It is a long-standing problem in graph theory to prove or disprove the \emph{reconstruction conjecture}, also known as the Kelly-Ulam conjecture. This conjecture states that every simple graph on at least three vertices is \emph{reconstructible}, which means that the isomorphism class of such a graph is uniquely determined by the isomorphism classes of its vertex-deleted subgraphs. In this talk, the notion of reconstructing is extended from graphs to instances of the constraint satisfaction problem (CSP): an instance is \emph{reconstructible} if its isomorphism class is uniquely determined by the isomorphism classes of its constraint-deleted subinstances. Questions of interest include not only questions about reconstructible CSP instances but also about CSP instances with reconstructible properties and parameters such as the existence of solutions or the number of solutions. As shown in the talk, such questions can be answered using techniques borrowed and adapted from graph reconstruction. In particular, Lov\'{a}sz's method of counting graph homomorphisms \cite{Lov72} is adapted to characterize CSP instances for which the number of solutions is reconstructible.

scholarly journals Kempe-Locking Configurations

2018 ◽  
Author(s):  
James A. Tilley
Keyword(s):  
Bipartite Graph ◽  
Systematic Search ◽  
Single Edge ◽  
Heuristic Argument ◽  
Bottom Boundary ◽  
Single Vertex ◽  
Planar Triangulation ◽  
Isomorphism Classes ◽  
Connectivity Property ◽  
Proper Subgraph

Existing proofs of the 4-color theorem succeeded by establishing an unavoidable set of reducible configurations. By this device, their authors showed that a minimum counterexample cannot exist. G.D. Birkhoff proved that a minimum counterexample must satisfy a connectivity property that is referred to in modern parlance as internal 6-connectivity. We show that a minimum counterexample must also satisfy a coloring property, one that we call Kempe-locking. We define the terms Kempe-locking configuration and fundamental Kempe-locking configuration. We provide a heuristic argument that a fundamental Kempe-locking configuration must be of low order and then perform a systematic search through isomorphism classes for such configurations. We describe a methodology for analyzing whether an arbitrary planar triangulation is Kempe-locked; it involves deconstructing the triangulation into a stack of configurations with common endpoints and then creating a bipartite graph of coloring possibilities for each configuration in the stack to assess whether certain 2-color paths can be transmitted from the configuration's top boundary to its bottom boundary. All Kempe-locked triangulations we discovered have two features in common: (1) they are Kempe-locked with respect to only a single edge, say $xy$, and (2) they have a Birkhoff diamond with endpoints $x$ and $y$ as a proper subgraph. On the strength of our various investigations, we are led to a plausible conjecture that the Birkhoff diamond is the only fundamental Kempe-locking configuration. If true, this would establish that the connectivity and coloring properties of a minimum counterexample to the 4-color theorem are incompatible. It would also point to the singular importance of a particularly elegant 4-connected triangulation of order 9 that consists of a triangle enclosing a pentagon enclosing a single vertex.

A neigborhood union condition for nonadjacent vertices in graphs

2017 ◽  
Vol 09 (05) ◽  
pp. 1750060
Author(s):  
Farzad Shaveisi
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Simple Graph ◽  
Strongly Regular Graphs ◽  
Star Graph ◽  
Regular Graphs ◽  
Strongly Regular ◽  
Union Condition ◽  
Bounded Graph

A simple graph [Formula: see text] is called [Formula: see text]-bounded if for every two nonadjacent vertices [Formula: see text] of [Formula: see text] there exists a vertex [Formula: see text] such that [Formula: see text], where [Formula: see text] denotes the set of neighbors of the vertex [Formula: see text] in [Formula: see text]. In this paper, some properties of [Formula: see text]-bounded graphs are studied. It is shown that any bipartite [Formula: see text]-bounded graph is a complete bipartite graph with at most two horns; in particular, any [Formula: see text]-bounded tree is either a star or a two-star graph. Also, we prove that any non-end vertex of every [Formula: see text]-bounded graph is contained in either a triangle or a rectangle. Among other results, it is shown that all regular [Formula: see text]-bounded graphs are strongly regular graphs. Finally, we determine that how many edges can an [Formula: see text]-bounded graph have?

scholarly journals On the maximum ABC index of bipartite graphs without pendent vertices

2020 ◽  
Vol 18 (1) ◽  
pp. 39-49 ◽  
Author(s):  
Zehui Shao ◽  
Pu Wu ◽  
Huiqin Jiang ◽  
S.M. Sheikholeslami ◽  
Shaohui Wang
Keyword(s):  
Bipartite Graph ◽  
Bipartite Graphs ◽  
Simple Graph ◽  
Connectivity Index ◽  
Degree Of Vertex ◽  
Abc Index ◽  
Pendent Vertices ◽  
Atom Bond

AbstractFor a simple graph G, the atom–bond connectivity index (ABC) of G is defined as ABC(G) = $\sum_{uv\in{}E(G)} \sqrt{\frac{d(u)+d(v)-2}{d(u)d(v)}},$where d(v) denotes the degree of vertex v of G. In this paper, we prove that for any bipartite graph G of order n ≥ 6, size 2n − 3 with δ(G) ≥ 2, $ABC(G)\leq{}\sqrt{2}(n-6)+2\sqrt{\frac{3(n-2)}{n-3}}+2,$and we characterize all extreme bipartite graphs.

THE SECOND SKEW-SYMMETRIC COHOMOLOGY GROUP AND SPATIAL EMBEDDINGS OF GRAPHS

2000 ◽  
Vol 09 (03) ◽  
pp. 387-411 ◽  
Author(s):  
RYO NIKKUNI
Keyword(s):  
Bipartite Graph ◽  
Equivalence Relation ◽  
Complete Graph ◽  
Cohomology Group ◽  
Complete Bipartite Graph ◽  
Simple Graph ◽  
Residual Space ◽  
Symmetric Cohomology

Let L(G) be the second skew-symmetric cohomology group of the residual space of a graph G. We determine L(G) in the case G is a 3-connected simple graph, and give the structure of L(G) in the case of G is a complete graph and a complete bipartite graph. By using these results, we determine the Wu invariants in L(G) of the spatial embeddings of the complete graph and those of the complete bipartite graph, respectively. Since the Wu invariant of a spatial embedding is a complete invariant up to homology which is an equivalence relation on spatial embeddings introduced in [12], we give a homology classification of the spatial embeddings of such graphs.

scholarly journals Partitioning to three matchings of given size is NP-complete for bipartite graphs

2014 ◽  
Vol 6 (2) ◽  
pp. 206-209 ◽  
Author(s):  
Dömötör Pálvölgyi
Keyword(s):  
Bipartite Graph ◽  
Maximum Degree ◽  
Perfect Matching ◽  
Bipartite Graphs ◽  
Simple Graph ◽  
Np Complete

Abstract We show that the problem of deciding whether the edge set of a bipartite graph can be partitioned into three matchings, of size k1, k2 and k3 is NP-complete, even if one of the matchings is required to be perfect. We also show that the problem of deciding whether the edge set of a simple graph contains a perfect matching and a disjoint matching of size k or not is NP-complete, already for bipartite graphs with maximum degree 3. It also follows from our construction that it is NP-complete to decide whether in a bipartite graph there is a perfect matching and a disjoint matching that covers all vertices whose degree is at least 2.

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