Simple Bipartite Graph Research Articles

REGULARITY OF POWERS OF BIPARTITE GRAPHS

AbstractFor a simple bipartite graph G, we give an upper bound for the regularity of powers of the edge ideal $I(G)$ in terms of its vertex domination number. Consequently, we explicitly compute the regularity of powers of the edge ideal of a bipartite Kneser graph. Further, we compute the induced matching number of a bipartite Kneser graph.

The sum necessary to ensure that a degree sequence pair has an [formula omitted]-connected realization

Let S=(a1,…,am;b1,…,bn), where a1,…,am and b1,…,bn are two nonincreasing sequences of nonnegative integers. The pair S=(a1,…,am;b1,…,bn) is said to be a bigraphic pair if there is a simple bipartite graph G=(X∪Y,E) such that a1,…,am and b1,…,bn are the degrees of the vertices in X and Y, respectively. Let A be an (additive) Abelian group. We define σ(A,m,n) to be the minimum integer k such that every bigraphic pair S=(a1,…,am;b1,…,bn) with am,bn≥2 and σ(S)=a1+⋯+am≥k has an A-connected realization. In this paper, we determine the values of σ(A,m,n) for |A|=k and m≥n≥2, where k≥6 is an even integer, and the values of σ(A,m,n) for |A|=4 and m≥n≥3. Therefore, together with some previous results [(Yin, 2016); (Yin and Dai, 2017); (Guan and Yin, 2018)], the values of σ(A,m,n) for |A|≥3 are determined completely.

Catlin’s reduced graphs with small orders

A graph is supereulerian if it has a spanning closed trail. Catlin in 1990 raised the problem of determining the reduced nonsupereulerian graphs with small orders, as such results are of particular importance in the study of Eulerian subgraphs and Hamiltonian line graphs. We determine all reduced graphs with order at most 14 and with few vertices of degree 2, extending former results of Chen and Chen in 2016. In 1985, Bauer proposed the problems of determining best possible sufficient conditions on minimum degree of a simple graph (or a simple bipartite graph, respectively) G to ensure that its line graph L(G) is Hamiltonian. These problems have been settled by Catlin and Lai in 1988, respectively. As an application of our main results, we prove the following for a connected simple graph G on n vertices: If then for sufficiently large n, L(G) is Hamilton-connected if and only if both and G is not nontrivially contractible to the Wagner graph. If G is bipartite and then for sufficiently large n, L(G) is Hamilton-connected if and only if both and G is not nontrivially contractible to the Wagner graph.

On Finding Bipartite Graphs With a Small Number of Short Cycles and Large Girth

The problem of finding bipartite (Tanner) graphs with given degree sequences that have large girth and few short cycles is of great interest in many applications including construction of good low-density parity-check (LDPC) codes. In this paper, we prove that for given integers $\alpha $ , $\beta $ , and $\gamma $ , and degree sequences $\pi $ and $\pi '$ , the problem of determining whether there exists a simple bipartite graph with degree sequences $(\pi, \pi ')$ that has at most $\alpha $ ( $\beta $ and $\gamma $ ) cycles of length four (six and eight, respectively) is NP-complete. This is proved by a two-step polynomial-time reduction from the 3-Partition Problem. On the other hand, using connections to linear hypergraphs, we prove that given the degree sequence $\pi $ , a polynomial time algorithm can be devised to determine whether there exists a bipartite graph whose degree sequence on one side of the bipartition is $\pi $ and has a girth of at least six. In addition to these complexity results, we devise a quasi-polynomial time algorithm that can construct a bipartite graph with given (irregular) degree sequences and a girth that increases logarithmically with the size of the graph. Compared to the well-known progressive-edge-growth (PEG) algorithm, the proposed method, in some cases, results in Tanner graphs with larger girth, particularly for scenarios where the degree sequences of the graph are strictly enforced.

Existence of non-Cayley Haar graphs

A Cayley graph of a group H is a finite simple graph Γ such that its automorphism group Aut(Γ) contains a subgroup isomorphic to H acting regularly on V(Γ), while a Haar graph of H is a finite simple bipartite graph Σ such that Aut(Σ) contains a subgroup isomorphic to H acting semiregularly on V(Σ) and the H-orbits are equal to the partite sets of Σ. It is well-known that every Haar graph of finite abelian groups is a Cayley graph. In this paper, we prove that every finite non-abelian group admits a non-Cayley Haar graph except the dihedral groups D6, D8, D10, the quaternion group Q8 and the group Q8×Z2. This answers an open problem proposed by Estélyi and Pisanski in 2016.

Resonance graphs of catacondensed even ring systems

A catacondensed even ring system (shortly CERS) is a simple bipartite 2-connected outerplanar graph with all vertices of degree 2 or 3. In this paper, we investigate the resonance graphs (also called Z-transformation graphs) of CERS and firstly show that two even ring chains are evenly homeomorphic iff their resonance graphs are isomorphic. As the main result, we characterize CERS whose resonance graphs are daisy cubes. In this way, we greatly generalize the result known for kinky benzenoid graphs. Finally, some open problems are also presented.

On the sizes of bi-k-maximal graphs

Let $$k,n, s, t > 0$$ be integers and $$n = s+t \ge 2k+2$$. A simple bipartite graph G spanning $$K_{s,t}$$ is bi-k-maximal, if every subgraph of G has edge-connectivity at most k but any edge addition that does not break its bipartiteness creates a subgraph with connectivity at least $$k+1$$. We investigate the optimal size bounds of the bi-k-maximal simple graphs, and prove that if G is a bi-k-maximal graph with $$\min \{s, t \} \ge k$$ on n vertices, then each of the following holds.

On groups all of whose Haar graphs are Cayley graphs

A Cayley graph of a group $H$ is a finite simple graph $\Gamma$ such that ${\rm Aut}(\Gamma)$ contains a subgroup isomorphic to $H$ acting regularly on $V(\Gamma)$, while a Haar graph of $H$ is a finite simple bipartite graph $\Sigma$ such that ${\rm Aut}(\Sigma)$ contains a subgroup isomorphic to $H$ acting semiregularly on $V(\Sigma)$ and the $H$-orbits are equal to the bipartite sets of $\Sigma$. A Cayley graph is a Haar graph exactly when it is bipartite, but no simple condition is known for a Haar graph to be a Cayley graph. In this paper, we show that the groups $D_6, \, D_8, \, D_{10}$ and $Q_8$ are the only finite inner abelian groups all of whose Haar graphs are Cayley graphs (a group is called inner abelian if it is non-abelian, but all of its proper subgroups are abelian). As an application, it is also shown that every non-solvable group has a Haar graph which is not a Cayley graph.

English

Let Ks,t be the complete bipartite graph with partite sets {x1, …, xs} and {y1, …, yt}. A split bipartite-graph on (s + s′) + (t + t′) vertices, denoted by SBs + s′, t + t′, is the graph obtained from Ks,t by adding s′ + t′ new vertices xs + 1, …, xs + s′}, yt + 1, …, yt + t′ such that each of xs + 1, …, xs + s′; is adjacent to each of y1, …, yt and each of yt + 1, …, yt + t′ is adjacent to each of x1, …, xs. Let A and B be nonincreasing lists of nonnegative integers, having lengths m and n, respectively. The pair (A; B) is potentially SBs + s′, t + t′-bigraphic if there is a simple bipartite graph containing SBs + s′, t + t′ (with s + s′ vertices x1, …, xs + s′ in the part of size m and t + t′ vertices y1, …, yt + t′ in the part of size n) such that the lists of vertex degrees in the two partite sets are A and B. In this paper, we give a characterization for (A; B) to be potentially SBs + s′, t + t′-bigraphic. A simplification of this characterization is also presented.

Rejection sampling of bipartite graphs with given degree sequence

Abstract Let A = (a1, a2, ..., an) be a degree sequence of a simple bipartite graph. We present an algorithm that takes A as input, and outputs a simple bipartite realization of A, without stalling. The running time of the algorithm is ⊝(n1n2), where ni is the number of vertices in the part i of the bipartite graph. Then we couple the generation algorithm with a rejection sampling scheme to generate a simple realization of A uniformly at random. The best algorithm we know is the implicit one due to Bayati, Kim and Saberi (2010) that has a running time of O(mamax), where m = 1 2 ∑ i = 1 n a i $m = {1 \over 2}\sum\nolimits_{i = 1}^n {{a_i}} and amax is the maximum of the degrees, but does not sample uniformly. Similarly, the algorithm presented by Chen et al. (2005) does not sample uniformly, but nearly uniformly. The realization of A output by our algorithm may be a start point for the edge-swapping Markov Chains pioneered by Brualdi (1980) and Kannan et al.(1999).

Degree sum and hamiltonian-connected line graphs

In 1984, Bauer proposed the problems of determining best possible sufficient conditions on the vertex degrees of a simple graph (or a simple bipartite graph, or a simple triangle-free graph, respectively) G to ensure that its line graph L(G) is hamiltonian. We investigate the problems of determining best possible sufficient conditions on the vertex degrees of a simple graph G to ensure that its line graph L(G) is hamiltonian-connected, and prove the following.(i) For any real numbers a,b with 0<a<1, there exists a finite family F(a,b) such that for any connected simple graph G on n vertices, if dG(u)+dG(v)≥an+b for any u,v∈V(G) with uv∉E(G), then either L(G) is hamiltonian-connected, or κ(L(G))≤2, or L(G) is not hamiltonian-connected, κ(L(G))≥3 and G is contractible to a member in F(a,b).(ii) Let G be a connected simple graph on n vertices. If dG(u)+dG(v)≥n4−2 for any u,v∈V(G) with uv∉E(G), then for sufficiently large n, either L(G) is hamiltonian-connected, or κ(L(G))≤2, or L(G) is not hamiltonian-connected, κ(L(G))≥3 and G is contractible to W8, the Wagner graph.(iii) Let G be a connected simple triangle-free (or bipartite) graph on n vertices. If dG(u)+dG(v)≥n8 for any u,v∈V(G) with uv∉E(G), then for sufficiently large n, either L(G) is hamiltonian-connected, or κ(L(G))≤2, or L(G) is not hamiltonian-connected, κ(L(G))≥3 and G is contractible to W8, the Wagner graph.

Bigraphic pairs with an [formula omitted]-connected realization

Let S=(a1,…,am;b1,…,bn), where a1,…,am and b1,…,bn are two nonincreasing sequences of nonnegative integers. The pair S=(a1,…,am;b1,…,bn) is said to be a bigraphic pair if there is a simple bipartite graph G=(X∪Y,E) such that a1,…,am and b1,…,bn are the degrees of the vertices in X and Y, respectively. Let A be an (additive) Abelian group. We define σ(A,m,n) to be the minimum integer k such that every bigraphic pair S=(a1,…,am;b1,…,bn) with am,bn≥2 and σ(S)=a1+⋯+am≥k has an A-connected realization. In this paper, we determine the values of σ(A,m,n) for |A|=k and m≥n≥2, where k≥5 is an odd integer.

Using conics to construct geometric 3-configurations, part II: the generalized Steiner construction

In a number of previous papers, techniques have been developed to construct geometric 3-configurations whose reduced Levi graph has at least one pair of parallel edges, but construction techniques for configurations whose reduced Levi graph is a simple bipartite cubic graph remained elusive. In Using conics to construct geometric 3-configurations, part I: symmetrically generalizing the Pappus configuration (2016), a primarily straightedge-and-compass construction, along with the construction of a certain conic section, was developed to construct configurations whose reduced Levi graph is an edge-labelled digraph with underlying graph \(K_{3,3}\). In this paper, we extend that construction to produce a construction technique for certain classes of configurations whose reduced Levi graphs are bipartite simple cubic graphs that contain at least one quadrilateral; in particular, we produce a geometric construction method that produces configurations whose reduced Levi graphs have as their underlying graphs the cyclic fence graphs, which unify even prism graphs and odd Moebius ladders.

Solution to an extremal problem on bigraphic pairs with a Z 3-connected realization

Let S = (a 1, …, a m ; b 1, …, b n ), where a 1, …, a m and b 1, …, b n are two nonincreasing sequences of nonnegative integers. The pair S = (a 1, …, a m ; b 1, …, b n ) is said to be a bigraphic pair if there is a simple bipartite graph G = (X ∪ Y, E) such that a 1, …, a m and b 1, …, b n are the degrees of the vertices in X and Y, respectively. Let Z 3 be the cyclic group of order 3. Define σ(Z 3, m, n) to be the minimum integer k such that every bigraphic pair S = (a 1, …, a m ; b 1, …, b n ) with a m , b n ≥ 2 and σ(S) = a 1 + ⋯ + a m ≥ k has a Z 3-connected realization. For n = m, Yin [Discrete Math., 339, 2018—2026 (2016)] recently determined the values of σ(Z 3, m, m) for m ≥ 4. In this paper, we completely determine the values of σ(Z 3, m, n) for m ≥ n ≥ 4.

An extension of a result of Alon, Ben-Shimon and Krivelevich on bipartite graph vertex sequences

Let S=(a1,…,am;b1,…,bn) be a pair of two nonincreasing sequences of positive integers, that is, a1,…,am and b1,…,bn are two nonincreasing sequences of positive integers. The pair S=(a1,…,am;b1,…,bn) is said to be a bigraphic pair if there is a simple bipartite graph G=(X∪Y,E) such that a1,…,am and b1,…,bn are the degrees of the vertices in X and Y, respectively. In this paper, we give a simple sufficient condition for a pair S=(a1,…,am;b1,…,bn) to be a bigraphic pair. The condition depends only on the lengths of two sequences of the pair and their largest and smallest elements. This result extends a result due to Alon et al. (2010).

Induced forests in bipartite planar graphs

Akiyama and Watanabe conjectured that every simple planar bipartite graph on $n$ vertices contains an induced forest on at least $5n/8$ vertices. We apply the discharging method to show that every simple bipartite planar graph on $n$ vertices contains an induced forest on at least $\lceil (4n+3)/7 \rceil$ vertices.

The 3-flow conjecture, factors modulo k, and the 1-2-3-conjecture

Let k be an odd natural number ≥5, and let G be a (6k−7)-edge-connected graph of bipartite index at least k−1. Then, for each mapping f:V(G)→N, G has a subgraph H such that each vertex v has H-degree f(v) modulo k. We apply this to prove that, if c:V(G)→Zk is a proper vertex-coloring of a graph G of chromatic number k≥5 or k−1≥6, then each edge of G can be assigned a weight 1 or 2 such that each weighted vertex-degree of G is congruent to c modulo k. Consequently, each nonbipartite (6k−7)-edge-connected graph of chromatic number at most k (where k is any odd natural number ≥3) has an edge-weighting with weights 1,2 such that neighboring vertices have distinct weighted degrees (even after reducing these weighted degrees modulo k). We characterize completely the bipartite graph having an edge-weighting with weights 1,2 such that neighboring vertices have distinct weighted degrees. In particular, that problem belongs to P while it is NP-complete for nonbipartite graphs. The characterization also implies that every 3-edge-connected bipartite graph with at least 3 vertices has such an edge-labeling, and so does every simple bipartite graph of minimum degree at least 3.

An extremal problem on bigraphic pairs with an [formula omitted]-connected realization

Let S=(a1,…,am;b1,…,bn), where a1,…,am and b1,…,bn are two nonincreasing sequences of nonnegative integers. The pair S=(a1,…,am;b1,…,bn) is said to be a bigraphic pair if there is a simple bipartite graph G=(X∪Y,E) such that a1,…,am and b1,…,bn are the degrees of the vertices in X and Y, respectively. Let A be an (additive) Abelian group. We define σ(A,m,n) to be the minimum integer k such that every bigraphic pair S=(a1,…,am;b1,…,bn) with am,bn≥2 and σ(S)=a1+⋯+am≥k has an A-connected realization. In this paper, we determine the values of σ(Z3,m,m) for m≥4.

A characterization of PM-compact Hamiltonian bipartite graphs

The perfect matching polytope of a graph G is the convex hull of the incidence vectors of all perfect matchings in G. A graph is called perfect matching compact (shortly, PM-compact), if its perfect matching polytope has diameter one. This paper gives a complete characterization of simple PM-compact Hamiltonian bipartite graphs. We first define two families of graphs, called the H2C-bipartite graphs and the H23-bipartite graphs, respectively. Then we show that, for a simple Hamiltonian bipartite graph G with |V(G)| ≥ 6, G is PM-compact if and only if G is K3,3, or G is a spanning Hamiltonian subgraph of either an H2C-bipartite graph or an H23-bipartite graph.

Spectral radius of bipartite graphs

Let k, p, q be positive integers with k<p<q+1. We prove that the maximum spectral radius of a simple bipartite graph obtained from the complete bipartite graph Kp,q of bipartition orders p and q by deleting k edges is attained when the deleted edges are all incident on a common vertex which is located in the partite set of order q. Our method is based on new sharp upper bounds on the spectral radius of bipartite graphs in terms of their degree sequences.