Balanced Bipartite Graphs Research Articles

Measuring associativity: graph algebras of undirected graphs

We study two measures of associativity for graph algebras of finite undirected graphs: the index of nonassociativity and (a variant of) the semigroup distance. We determine “almost associative” and “antiassociative” graphs with respect to both measures. It turns out that the antiassociative graphs are exactly the balanced complete bipartite graphs, no matter which of the two measures we consider. In the class of connected graphs the two notions of almost associativity are also equivalent.

2-Factors with a Few Components in Balanced Bipartite Graphs

2-Factors with a Few Components in Balanced Bipartite Graphs

A Sufficient Condition for Vertex Bipancyclicity in Balanced Bipartite Graphs

A Sufficient Condition for Vertex Bipancyclicity in Balanced Bipartite Graphs

Extremal bipartite independence number and balanced coloring

In this paper, we establish a couple of results on extremal problems in bipartite graphs. Firstly, we show that every sufficiently large bipartite graph with average degree D and with n vertices on each side has a balanced independent set containing (1−ϵ)logDDn vertices from each side for small ϵ>0. Secondly, we prove that the vertex set of every sufficiently large balanced bipartite graph with maximum degree at most Δ can be partitioned into (1+ϵ)ΔlogΔ balanced independent sets. Both of these results are algorithmic and best possible up to a factor of 2, which might be hard to improve as evidenced by the phenomenon known as ‘algorithmic barrier’ in the literature. The first result improves a recent theorem of Axenovich, Sereni, Snyder, and Weber in a slightly more general setting. The second result improves a theorem of Feige and Kogan about coloring balanced bipartite graphs.

The Rank of the Sandpile Group of Random Directed Bipartite Graphs

We identify the asymptotic distribution of p-rank of the sandpile group of random directed bipartite graphs which are not too imbalanced. We show this matches exactly with that of the Erdös–Rényi random directed graph model, suggesting that the Sylow p-subgroups of this model may also be Cohen–Lenstra distributed. Our work builds on the results of Koplewitz who studied p-rank distributions for unbalanced random bipartite graphs, and showed that for sufficiently unbalanced graphs, the distribution of p-rank differs from the Cohen–Lenstra distribution. Koplewitz (sandpile groups of random bipartite graphs, https://arxiv.org/abs/1705.07519, 2017) conjectured that for random balanced bipartite graphs, the expected value of p-rank is O(1) for any p. This work proves his conjecture and gives the exact distribution for the subclass of directed graphs.

Degree sum condition on distance 2 vertices for hamiltonian cycles in balanced bipartite graphs

Let G be a 2-connected balanced bipartite graph of order 2n with n≥3. Denote μ(G)=min⁡{max⁡{d(x),d(y)}:x,y∈V(G),dist(x,y)=2} and μ2(G)=min⁡{d(x)+d(y):x,y∈V(G),dist(x,y)=2}. Wang and Liu (2018) [14] proved that if μ(G)≥n+12, then G is hamiltonian. In this paper, we characterize non-hamiltonian bipartite graphs with μ(G)=n2. Also, we show that G is bipancyclic with one exception, if μ(G)≥n+12. As a direct consequence, we also show that if μ2(G)≥n+1, then G is hamiltonian and, with one exception, G is bipancyclic. Moreover, if we replace n+1 with n+2, then G contains a hamiltonian cycle passing through every edge of a perfect matching. The lower bounds in all results are sharp.

Diameter Three Orientability of Bipartite Graphs

In 2019, Czabarka, Dankelmann and Székely showed that for every undirected graph of order $n$, the minimum degree threshold for diameter two orientability is $\frac{n}{2}+ \Theta(\ln n)$. In this paper, we consider bipartite graphs and give a sufficient condition in terms of the minimum degree for such graphs to have oriented diameter three. We in particular prove that for balanced bipartite graphs of order $n$, the minimum degree threshold for diameter three orientability is $\frac{n}{4}+\Theta(\ln n)$.

On some applications related with algebraic structures through different well known graphs

Lot of achievements have been done to comprehend the latest phenomena of our life by discussing the interrelationship of two boughs, graph theory and algebra, of mathematics. In the modern time, it seems to be of great importance to study about eminent, extensive and certifiable applications through well known tools of graph theory. Combinatorial commutative algebra, moderately new, quickly creating numerical tool and foremost tendril of mathematics, has been used exhaustively to solve a list of open problems in algebra under the shadow of graphs. It can be helpful, instead of algebraically, to find the metric dimension of zero-divisor graphs, NP-complete problem, related with finite mathematical structures. In this article we shall discuss some graphs, zero-divisor graphs of finite Boolean rings and balanced bipartite graphs of conjugacy closed loops, and their associated applications.

Biholes in balanced bipartite graphs

A bihole in a bipartite graph $G$ with partite sets $A$ and $B$ is an independent set $I$ in $G$ with $|I\cap A|=|I\cap B|$. We prove lower bounds on the largest order of biholes in balanced bipartite graphs subject to conditions involving the vertex degrees and the average degree.

Unified Spectral Hamiltonian Results of Balanced Bipartite Graphs and Complementary Graphs

There have been researches on sufficient spectral conditions for Hamiltonian properties and path-coverable properties of graphs. Utilizing the Bondy–Chvatal closure, we provide a unified approach to study sufficient graph eigenvalue conditions for these properties and sharpen several former spectral Hamiltonian results on balanced bipartite graphs and complementary graphs.

Some generalizations of spectral conditions for 2s-hamiltonicity and 2s-traceability of bipartite graphs

ABSTRACT This paper mainly focuses on spectral conditions for the Hamiltonicity of balanced bipartite graphs and nearly balanced bipartite graphs. Let and be the spectral radius and signless Laplacian spectral radius of a graph G, respectively. One of our main results is the following theorem: Let G be a balanced bipartite graph of order and of minimum degree . If and , then G is -hamiltonian unless . If and , then G is -hamiltonian unless . This theorem generalizes the results in [B. L. Li, B. Ning, Spectral analogues of Erdos' and Moon–Moser's theorems on Hamilton cycles, Linear and Multilinear Algebra, 64 (2016) 2252–2269].

Sufficient conditions for graphs to be spanning connected

A graph G is t*-connected if there exist t internally disjoint (u, v)-paths, between any two vertices u and v, whose union spans G. In this sense, t*-connectedness is a natural extension of hamiltonicity. In this paper, we provide a sufficient condition for graphs to be t*-connected by generalizing a classic result given by Chavátal [7]. Furthermore, as byproducts, we extend some known results concerning fault tolerant hamiltonicity and minimum cardinality of edges. We also establish analogous results for balanced bipartite graphs.

Sufficient Conditions for Hamiltonicity of Graphs with Respect to Wiener Index, Hyper-Wiener Index, and Harary Index

In this paper, with respect to the Wiener index, hyper-Wiener index, and Harary index, it gives some sufficient conditions for some graphs to be traceable, Hamiltonian, Hamilton-connected, or traceable for every vertex. Firstly, we discuss balanced bipartite graphs with δG≥t, where δG is the minimum degree of G, and gain some sufficient conditions for the graphs to be traceable or Hamiltonian, respectively. Secondly, we discuss nearly balanced bipartite graphs with δG≥t and present some sufficient conditions for the graphs to be traceable. Thirdly, we discuss graphs with δG≥t and obtain some conditions for the graphs to be traceable or Hamiltonian, respectively. Finally, we discuss t-connected graphs and provide some conditions for the graphs to be Hamilton-connected or traceable for every vertex, respectively.

Spanning bipartite graphs with high degree sum in graphs

The classical Ore’s Theorem states that every graph G of order n≥3 with σ2(G)≥n is hamiltonian, where σ2(G)=min{dG(x)+dG(y):x,y∈V(G),x≠y,xy∉E(G)}. Recently, Ferrara, Jacobson and Powell (Discrete Math. 312 (2012), 459–461) extended the Moon–Moser Theorem and characterized the non-hamiltonian balanced bipartite graphs H of order 2n≥4 with partite sets X and Y satisfying σ1,1(H)≥n, where σ1,1(H)=min{dH(x)+dH(y):x∈X,y∈Y,xy∉E(H)}. Though the latter result apparently deals with a narrower class of graphs, we prove in this paper that it implies Ore’s Theorem for graphs of even order.

Cordial Labeling of Cartesian Product between two Balanced Bipartite Graphs

Cordial Labeling finds its application in Automated Routing algorithms, Communications relevant Adhoc Networks and many others. These networks and routing paths are best represented by a set of vertices and edges which forms a complex graph. In this paper an Algorithmic approach is devised in order to cordially label some of such complex graphs formed by the Cartesian product of two Balanced Bipartite Graphs. There is an absence of a universal algorithm that could label the entire family of \(K_{n,n} \times K_{n,n}\) graphs, which is primarily because of the complexity of these graphs. We have attempted to redefine the \(K_{n,n}\) graph so that on carrying out the Cartesian product we obtain a symmetrical graph. Subsequently we design an efficient and effective algorithm for cordially labeling \(G= K_{n,n} \times K_{n,n}\). We also proof that the algorithm is running with polynomial time complexity. The results can be used to study and design algorithms for signed product cordial, total signed product cordial, prime cordial labeling etc. of the family of \(K_{n,n} \times K_{n,n}\) graphs.

Spectral radius and Hamiltonian properties of graphs, II

In this paper, we first present spectral conditions for the existence of in graphs (2-connected graphs) of order n, which are motivated by a conjecture of Erds. We also prove spectral conditions for the existence of Hamilton cycles in balanced bipartite graphs. This result presents a spectral analog of Moon-Moser's theorem on Hamilton cycles in balanced bipartite graphs, and extends a previous theorem due to Li and the second author for n sufficiently large. We conclude this paper with two problems on tight spectral conditions for the existence of long cycles of given lengths.

Some new sufficient conditions for 2p-Hamilton-biconnectedness of graphs

A balanced bipartite graph G is said to be 2p-Hamilton-biconnected if for any balanced subset W of size 2p of V(G), the subgraph induced by V(G)nW is Hamilton-biconnected. In this paper, we prove that ?Let G be a balanced bipartite graph of order 2n with minimum degree ?(G) ? k, where n ? 2k-p+2 for two integers k ? p ? 0. If the number of edges e(G) > n(n-k + p-1) + (k + 2)(k-p+1), then G is 2p-Hamilton-biconnected except some exceptions.? Furthermore, this result is used to present two new spectral conditions for a graph to be 2p-Hamilton-biconnected. Moreover, the similar results are also presented for nearly balanced bipartite graphs.

Three-Color Bipartite Ramsey Number for Graphs with Small Bandwidth

We estimate the 3-color bipartite Ramsey number for balanced bipartite graphs $H$ with small bandwidth and bounded maximum degree. More precisely, we show that the minimum value of $N$ such that in...

Spectral radius and traceability of graphs with large minimum degree

ABSTRACTLet and be the spectral radius and the minimum degree of a graph G, respectively. In this paper, we obtain spectral conditions for a graph to be traceable among graphs (balanced bipartite graphs) with large minimum degree. The main results are listed as:Let G be a graph with order and , where is an integer. If then G is traceable unless or .Let G be a balanced bipartite graph with order and , where and . If then G is traceable unless , where is the graph obtained from by deleting.

Strong Geodetic Number of Complete Bipartite Graphs and of Graphs with Specified Diameter

The strong geodetic problem is a recent variation of the classical geodetic problem. For a graph G, its strong geodetic number $${{\mathrm{sg}}}(G)$$ is the cardinality of a smallest vertex subset S, such that each vertex of G lies on one fixed shortest path between a pair of vertices from S. In this paper, some general properties of the strong geodetic problem are studied, especially in connection with the diameter of a graph. The problem is also solved for balanced complete bipartite graphs.