Michael S. Lang, Paul M. Terwilliger
Let $G$ denote a $Q$-polynomial distance-regular graph with diameter $D$ at least 4. Assume that the intersection numbers of $G$ satisfy $a_i=0$ for $0 \leq i \leq D-1$ and $a_D\neq 0$. We show that $G$ is a polygon, a folded cube, or an Odd graph.
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Frankl's union-closed sets conjecture states that in every finite union-closed set of sets, there is an element that is contained in at least half of the member-sets (provided there are at least two members). The conjecture has an equivalent formulation in terms of graphs: In every bipartite graph with least one edge, both colour classes contain a vertex belonging to at most half of the maximal stable sets. We prove that, for every fixed edge-probability, almost every random bipartite graph almost satisfies Frankl's conjecture.
Let $Y$ denote a $D$-class symmetric association scheme with $D \geq 3$, and suppose $Y$ is almost-bipartite P- and Q-polynomial. Let $x$ denote a vertex of $Y$ and let $T=T(x)$ denote the corresponding Terwilliger algebra. We prove that any irreducible $T$-module $W$ is both thin and dual thin in the sense of Terwilliger. We produce two bases for $W$ and describe the action of $T$ on these bases. We prove that the isomorphism class of $W$ as a $T$-module is determined by two parameters, the dual endpoint and diameter of $W$. We find a recurrence which gives the multiplicities with which the i…
On Bipartite-Almost Bipartite Graphs and the Determinantal Factorization
本文提出BAB图概念，通过Gallai-Edmonds分解研究其结构，并证明其邻接矩阵行列式可分解，验证R-不相交图猜想。
In this paper we study almost Cohen-Macaulay bipartite graphs. Furthermore, we prove that if $G$ is almost Cohen-Macaulay bipartite graph with at least one vertex of positive degree, then there is a vertex of $°(v) \leq 2$. In particular, if $G$ is an almost Cohen-Macaulay bipartite graph and $u$ is a vertex of degree one of $G$ and $v$ its adjacent vertex, then $G\setminus\{v\}$ is almost Cohen-Macaulay. Also, we show that an unmixed Ferrers graph is almost Cohen-Macaulay if and only if it is connected in codimension two. Moreover, we give some examples.
We prove that for every $k$ and every $\varepsilon>0$, there exists $g$ such that every graph with tree-width at most $k$ and odd-girth at least $g$ has circular chromatic number at most $2+\varepsilon$.
In an effort to find more examples of amalgamation classes whose almost sure theories are the same as their generic theories as well as amalgamation classes whose almost sure theories are different from their generic theories, we address our attention to two new cases: the bipartite diameter 3 metrically homogeneous graphs of generic type. These graphs were identified by Cherlin, and are determined by certain forbidden configurations. In this paper, we explicitly identify and enumerate their ages, for which we then establish both unlabeled and labeled 0-1 laws. Finally, we show that for one of…
The neighborhood of a pair of vertices $u,v$ in a triple system is the set of vertices $w$ such that $uvw$ is an edge. A triple system $\HH$ is semi-bipartite if its vertex set contains a vertex subset $X$ such that every edge of $\HH$ intersects $X$ in exactly two points. It is easy to see that if $\HH$ is semi-bipartite, then the neighborhood of every pair of vertices in $\HH$ is an independent set. We show a partial converse of this statement by proving that almost all triple systems with vertex sets $[n]$ and independent neighborhoods are semi-bipartite. Our result can be viewed as an exte…
Let $G$ be a bipartite graph without loops and multiple edges on $v\ge 4$ vertices, which can be drawn on the plane such that any edge intersects at most one other edge. We prove that such graph has at most $3v-8$ edges for even $v\ne 6$ and at most $3v-9$ edges for odd $v$ and $v=6$. For all $v\ge 4$ examples showing that these bounds are tight are constructed. In the end of paper we discuss a question about drawings of complete bipartite graphs on the plane such that any edge intersects at most one other edge. {\sc Keywords:} topological graphs, planar graphs, bipartite graphs.
A regular bipartite tournament is an orientation of a complete balanced bipartite graph $K_{2n,2n}$ where every vertex has its in- and outdegree both equal to $n$. In 1981, Jackson conjectured that any regular bipartite tournament can be decomposed into Hamilton cycles. We prove this conjecture for all sufficiently large bipartite tournaments. Along the way, we also prove several further results, including a conjecture of Liebenau and Pehova on Hamilton decompositions of dense bipartite digraphs.
A hypermap is bipartite if its set of flags can be divided into two parts A and B so that both A and B are the union of vertices, and consecutive vertices around an edge or a face are contained in alternate parts. A bipartite hypermap is bipartite-regular if its set of automorphisms is transitive on A and on B. In this paper we see some properties of the constructions of bipartite hypermaps described algebraically by Breda and Duarte which generalize the construction induced by the Walsh representation of hypermaps. As an application we show that all surfaces have bipartite-regular hypermaps.
Gyárfas proved that every coloring of the edges of $K_n$ with $t+1$ colors contains a monochromatic connected component of size at least $n/t$. Later, Gyárfás and Sárközy asked for which values of $γ=γ(t)$ does the following strengthening for almost complete graphs hold: if $G$ is an $n$-vertex graph with minimum degree at least $(1-γ)n$, then every $(t+1)$-edge coloring of $G$ contains a monochromatic component of size at least $n/t$. We show $γ= 1/(6t^3)$ suffices, improving a result of DeBiasio, Krueger, and Sárközy.
In this article, we define a family of regular bipartite graphs and show that the homotopy type of the independence complexes of this family is the wedge sum of spheres of certain dimensions.
On R-disjoint graphs: a generalization of almost bipartite non-König-Egerváry graphs
本文提出R-不相交图的概念，推广了非康尼格-埃格瓦里几乎二分图的理论，证明了其基本性质并修正了相关公式。
Let G be a simple balanced bipartite graph on $2n$ vertices, $δ= δ(G)/n$, and $ρ={δ+ \sqrt{2 δ-1} \over 2}$. If $δ> 1/2$ then it has a $\lfloor ρn \rfloor$-regular spanning subgraph. The statement is nearly tight.
It is shown that a hamiltonian $n/2$-regular bipartite graph $G$ of order $2n>8$ contains a cycle of length $2n-2$. Moreover, if such a cycle can be chosen to omit a pair of adjacent vertices, then $G$ is bipancyclic.
In this paper we study the local law for eigenvalues of large random regular bipartite graphs with degree growing arbitrarily fast. We prove that the empirical spectral distribution of the adjacency matrix converges to a scaled down copy of the Marchenko - Pastur distribution on intervals of short length.
We study bipartite distance-regular Cayley graphs with diameter three or four. We give sufficient conditions under which a bipartite Cayley graph can be constructed on the semidirect product of a group -- the part of this bipartite Cayley graph which contains the identity element -- and $\mathbb{Z}_{2}$. We apply this to the case of bipartite distance-regular Cayley graphs with diameter three, and consider cases where the sufficient conditions are not satisfied for some specific groups such as the dihedral group. We also extend a result by Miklavič and Potočnik that relates difference sets to …
In this paper, we show that every bipartite distance-regular Cayley graph with diameter $3$ can be constructed on the semidirect product of a group and $\mathbb{Z}_{2}$, except possibly for one case.
We show that for any $2$-local colouring of the edges of the balanced complete bipartite graph $K_{n,n}$, its vertices can be covered with at most~$3$ disjoint monochromatic paths. And, we can cover almost all vertices of any complete or balanced complete bipartite $r$-locally coloured graph with $O(r^2)$ disjoint monochromatic cycles.\\ We also determine the $2$-local bipartite Ramsey number of a path almost exactly: Every $2$-local colouring of the edges of $K_{n,n}$ contains a monochromatic path on $n$ vertices.
We consider a bipartite version of the color degree matrix problem. A bipartite graph $G(U,V,E)$ is half-regular if all vertices in $U$ have the same degree. We give necessary and sufficient conditions for a bipartite degree matrix (also known as demand matrix) to be the color degree matrix of an edge-disjoint union of half-regular graphs. We also give necessary and sufficient perturbations to transform realizations of a half-regular degree matrix into each other. Based on these perturbations, a Markov chain Monte Carlo method is designed in which the inverse of the acceptance ratios are polyn…
Kempe equivalence of almost bipartite graphs
本文研究几乎二部图的Kempe等价性问题，提出相关猜想并证明若干部分结果，同时指出该猜想后被Cranston和Feghali证明在一般情况下不成立。
The graph packing problem is a well-known area in graph theory. We consider a bipartite version and give almost tight conditions on the packability of two bipartite sequences.
We determine connected bipartite regular graphs with four distinct adjacency eigenvalues that induce periodic Grover walks, and show that it is only $C_6$. We also show that there are only three kinds of the second largest eigenvalues of bipartite regular periodic graphs with five distinct eigenvalues. Using walk-regularity, we enumerate feasible spectra for such graphs.
Let $L_N = L_{MBM}(X_1,..., X_N; Y_1,..., Y_N)$ be the minimum length of a bipartite matching between two sets of points in $\mathbf{R}^d$, where $X_1,..., X_N,...$ and $Y_1,..., Y_N,...$ are random points independently and uniformly distributed in $[0,1]^d$. We prove that for $d \ge 3$, $L_N/N^{1-1/d}$ converges with probability one to a constant $β_{MBM}(d)>0$ as $N\to \infty $.
In this paper, using matrix techniques, we compute the Ihara-zeta function and the number of spanning trees of the join of two semi-regular bipartite graphs. Furthermore, we show that the spectrum and the zeta function of the join of two semi-regular bipartite graphs can determine each other.
A 2-cell embedding of a graph $G$ into a closed (orientable or nonorientable) surface is called regular if its automorphism group acts regularly on the flags - mutually incident vertex-edge-face triples. In this paper, we classify the regular embeddings of complete bipartite graphs $K_{n,n}$ into nonorientable surfaces. Such regular embedding of $K_{n,n}$ exists only when $n = 2p_1^{a_1}p_2^{a_2}... p_k^{a_k}$ (a prime decomposition of $n$) and all $p_i \equiv \pm 1 (\mod 8)$. In this case, the number of those regular embeddings of $K_{n,n}$ up to isomorphism is $2^k$.
Consider the random process in which the edges of a graph $G$ are added one by one in a random order. A classical result states that if $G$ is the complete graph $K_{2n}$ or the complete bipartite graph $K_{n,n}$, then typically a perfect matching appears at the moment at which the last isolated vertex disappears. We extend this result to arbitrary $k$-regular bipartite graphs $G$ on $2n$ vertices for all $k = ω\left( \frac{n}{\log^{1/3} n} \right)$. Surprisingly, this is not the case for smaller values of $k$. Using a construction due to Goel, Kapralov and Khanna, we show that there exist bip…
In 2017, Qiao and Koolen showed that for any fixed integer $D\geq 3$, there are only finitely many such graphs with $θ_{\min}\leq -αk$, where $0<α<1$ is any fixed number. In this paper, we will study non-bipartite distance-regular graphs with relatively small $θ_{\min}$ compared with $k$. In particular, we will show that if $θ_{\min}$ is relatively close to $-k$, then the odd girth $g$ must be large. Also we will classify the non-bipartite distance-regular graphs with $θ_{\min} \leq \frac{D-1}{D}$ for $D =4,5$.
We introduce a new notation for representing labeled regular bipartite graphs of arbitrary degree. Several enumeration problems for labeled and unlabeled regular bipartite graphs have been introduced. A general algorithm for enumerating all non-isomorphic 2-regular bipartite graphs for a specified number of vertices has been described and a mathematical proof has been provided for its completeness. An abstraction of m Symmetric Permutation Tree in order to visualize a labeled r-Regular Bipartite Graph with 2m vertices and enumerate its automorphism group has been introduced. An algorithm to ge…
In this paper we are concerned with the classification of the finite groups admitting a bipartite DRR and a bipartite GRR. First, we find a natural obstruction in a finite group for not admitting a bipartite GRR. Then we give a complete classification of the finite groups satisfying this natural obstruction and hence not admitting a bipartite GRR. Based on these results and on some extensive computer computations, we state a conjecture aiming to give a complete classification of the finite groups admitting a bipartite GRR. Next, we prove the existence of bipartite DRRs for most of the finite g…
Let G be a bipartite graph with edge ideal I(G) whose quotient ring R/I(G) is sequentially Cohen-Macaulay. We prove: (1) the independence complex of G must be vertex decomposable, and (2) the Castelnuovo-Mumford regularity of R/I(G) can be determined from the invariants of G.
Almost-perfect colorful matchings in three-edge-colored bipartite graphs
该研究证明了三边着色二分图中存在满足各颜色边数指定条件的几乎完美彩色匹配，验证了Ryser-Brualdi-Stein猜想的三色多重扩展版本，且所得和的上界为最优。
For the set of graphs with a given degree sequence, consisting of any number of $2's$ and $1's$, and its subset of bipartite graphs, we characterize the optimal graphs who maximize and minimize the number of $m$-matchings. We find the expected value of the number of $m$-matchings of $r$-regular bipartite graphs on $2n$ vertices with respect to the two standard measures. We state and discuss the conjectured upper and lower bounds for $m$-matchings in $r$-regular bipartite graphs on $2n$ vertices, and their asymptotic versions for infinite $r$-regular bipartite graphs. We prove these conjectures…
We consider a bipartite distance-regular graph $G$ with diameter at least 4 and valency at least 3. Fix a vertex of $G$ and let $T$ denote the corresponding subconstituent algebra. We give a detailed description of a certain type of irreducible $T$-module, said to be thin with endpoint 2.
We study minimal free resolutions of edge ideals of bipartite graphs. We associate a directed graph to a bipartite graph whose edge ideal is unmixed, and give expressions for the regularity and the depth of the edge ideal in terms of invariants of the directed graph. For some classes of unmixed edge ideals, we show that the arithmetic rank of the ideal equals projective dimension of its quotient.
Regular and distance-regular characterizations of general graphs are well-known. In particular, the spectral excess theorem states that a connected graph G is distance-regular if and only if its spectral excess (a number that can be computed from the spectrum) equals the average excess (the mean of the numbers of vertices at extremal distance from every vertex). The aim of this paper is to derive new characterizations of regularity and distance-regularity for the more restricted family of bipartite graphs. In this case, some characterizations of (bi)regular bipartite graphs are given in terms …
When are random regular triangle-free graphs bipartite?
本文研究随机正则无三角图，发现其在 $d=\frac{\sqrt 3}{2}\sqrt{n \log n}$ 处发生尖锐相变，较小 $d$ 时非二部，较大 $d$ 时二部。
Characterizing bipartite distance-regularized graphs with vertices of eccentricity 4
本文证明特定参数与交数条件下的拟对称SPBIBD关联图，同$D=D'=4$的二分距离正则化图一一对应，从而刻画含离心率4顶点的图类。
Higher homological algebra for one-point extensions of bipartite hereditary algebras and spectral graph theory
该研究将2-表示有限二次单项式代数的高阶同调性质与对应二分图的谱图性质关联，分类了对应正则/边传递二分图的代数，并证明半正则对应二分图为自反图。
In this paper we are interested in the asymptotic enumeration of bipartite Cayley digraphs and Cayley graphs over abelian groups. Let $A$ be an abelian group and let $ι$ be the automorphism of $A$ defined by $a^ι=a^{-1}$, for every $a\in A$. A Cayley graph $\Cay(A, S)$ is said to have an automorphism group as small as possible if $\Aut(\Cay(A,S)) = \langle A,ι\rangle$. In this paper, we show that, except for two infinite families, almost all bipartite Cayley graphs on abelian groups have automorphism group as small as possible. We also investigate the analogous question for bipartite Cayley di…
The regular embeddings of complete bipartite graphs $K_{n,n}$ in orientable surfaces are classified and enumerated, and their automorphism groups and combinatorial properties are determined. The method depends on earlier classifications in the cases where $n$ is a prime power, obtained in collaboration with Du, Kwak, Nedela and Škoviera, together with results of Itô, Hall, Huppert and Wielandt on factorisable groups and on finite solvable groups.
Counting independent sets in expanding bipartite regular graphs
本文针对满足顶点扩张性质的一般正则二部图，利用聚合物模型和簇展开给出独立集数量的渐近展开，并证明其涵盖超立方体、中间层图及有界二部正则基图的笛卡尔积，同时提供顶点扩张的强界。
Recently, working on the Tanner graph which represents a low density parity check (LDPC) code becomes an interesting research subject. Finding the number of short cycles of Tanner graphs motivated Blake and Lin to investigate the multiplicity of cycles of length girth in bi-regular bipartite graphs, by using the spectrum and degree distribution of the graph. Although there were many algorithms to find the number of cycles, they preferred to investigate in a computational way. Dehghan and Banihashemi counted the number of cycles of length $g+2$ and $g+4,$ where $G$ is a bi-regular bipartite gra…
A graph has a perfect partition if all its perfect matchings can be partitioned so that each part is a 1-factorization of the graph. Let $L_{rm, r}=K_{rm,rm}-mK_{r,r}$. We first give a formula to count the number of perfect matchings of $L_{rm, r}$, then show that $L_{6,1}$ and $L_{8,2}$ have perfect partitions.
Let $\mathrm{rex}(n, F)$ denote the maximum number of edges in an $n$-vertex graph that is regular and does not contain $F$ as a subgraph. We give lower bounds on $\mathrm{rex}(n, F)$, that are best possible up to a constant factor, when $F$ is one of $C_4$, $K_{2,t}$, $K_{3,3}$ or $K_{s,t}$ when $t>s!$.
Sampling from the antiferromagnetic Ising model on bipartite, regular expander graphs
本文研究了在双部正则扩张图上反铁磁伊辛模型的采样问题，发现Glauber动力学混合缓慢，但提出高效算法和配分函数近似方法。
Kempe Classes and Almost Bipartite Graphs
本文研究近似二部图的 k-着色 Kempe 等价类，给出保证唯一等价类的新增边数阈值，并构造反例证明其渐近紧、反驳相关猜想。
Let $D$ be a $k$-regular bipartite tournament on $n$ vertices. We show that, for every $p$ with $2 \le p \le n/2-2$, $D$ has a cycle $C$ of length $2p$ such that $D \setminus C$ is hamiltonian unless $D$ is isomorphic to the special digraph $F_{4k}$. This statement was conjectured by Manoussakis, Song and Zhang [K. Zhang, Y. Manoussakis, and Z. Song. Complementary cycles containing a fixed arc in diregular bipartite tournaments. Discrete Mathematics, 133(1-3):325--328,1994]. In the same paper, the conjecture was proved for $p=2$ and more recently Bai, Li and He gave a proof for $p=3$ [Y. Bai, …
A set of vertices $W$ resolves a graph $G$ if every vertex is uniquely determined by its vector of distances to the vertices in $W$. A metric dimension of $G$ is the minimum cardinality of a resolving set of $G$. A bipartite graph G(n,n) is a graph whose vertex set $V$ can be partitioned into two subsets $V_1$ and $V_2,$ with $|V_1|=|V_2|=n,$ such that every edge of $G$ joins $V_1$ and $V_2$. The graph $G$ is called $k$-regular if every vertex of $G$ is adjacent to $k$ other vertices. In this paper, we determine the metric dimension of $k$-regular bipartite graphs G(n,n) where $k=n-1$ or $k=n-…
Flow polytopes for extensions of bipartite graphs
本文针对最小度至少为2的二分图，构造其扩展图和几乎度胡须图，证明扩展图的流多胞体的归一化体积等于胡须图中匹配数，并细化得到Ehrhart $h^*$-多项式等于无符号匹配多项式。
A labeling of a graph is a bijection from $E(G)$ to the set $\{1, 2,..., |E(G)|\}$. A labeling is \textit{antimagic} if for any distinct vertices $u$ and $v$, the sum of the labels on edges incident to $u$ is different from the sum of the labels on edges incident to $v$. We say a graph is antimagic if it has an antimagic labeling. In 1990, Ringel conjectured that every connected graph other than $K_2$ is antimagic. In this paper, we show that every regular bipartite graph (with degree at least 2) is antimagic. Our technique relies heavily on the Marriage Theorem.
We study the Castelnuovo-Mumford regularity of the vanishing ideal over a bipartite graph endowed with a decomposition of its edge set. We prove that, under certain conditions, the regularity of the vanishing ideal over a bipartite graph obtained from a graph by attaching a path of length $\ell$ increases by $\lfloor \frac{\ell}{2}\rfloor (q-2)$, where $q$ is the order of the field of coefficients. We use this result to show that the regularity of the vanishing ideal over a bipartite graph, $G$, endowed with a weak nested ear decomposition is equal to $$\textstyle \frac{|V_G|+ ε-3}{2}(q-2),$$ …
A hypermap is (hypervertex-) bipartite if its hypervertices can be 2-coloured in such a way that ``neighbouring'' hypervertices have different colours. It is bipartite-uniform if within each of the sets of hypervertices of the same colour, hyperedges and hyperfaces, elements have common valencies. The flags of a bipartite hypermap are naturally 2-coloured by assigning the colour of its adjacent hypervertices. A hypermap is bipartite-regular if the automorphism group acts transitively on each set of coloured flags. If the automorphism group acts transitively on the set of all flags, the hyperma…
Permutations avoiding bipartite partially ordered patterns have a regular insertion encoding
该研究证明避免高度≤2的偏序模式（POP）的排列类有正则插入编码及有理生成函数，还通过组合探索得到数百个避免大小为5的POP的排列类的组合规范与生成函数，解决了多个相关猜想。
Let $G$ be an abelian group of bounded exponent and $A \subseteq G$. We show that if the collection of translates of $A$ has VC dimension at most $d$, then for every $ε>0$ there is a subgroup $H$ of $G$ of index at most $ε^{-d-o(1)}$ such that one can add or delete at most $ε|G|$ elements to/from $A$ to make it a union of $H$-cosets. We also establish a removal lemma with polynomial bounds, with applications to property testing, for induced bipartite patterns in a finite abelian group with bounded exponent.
The independence number $α(G)$ is the cardinality of a maximum independent set, while $μ(G)$ is the size of a maximum matching in $G$. If $α(G)+μ(G)$ equals the order of $G$, then $G$ is called a Konig-Egervary graph. The number $d\left( G\right) =\max\{\left\vert A\right\vert -\left\vert N\left( A\right) \right\vert :A\subseteq V\}$ is called the critical difference of $G$ (where $N\left( A\right) =\left\{ v:v\in V,N\left( v\right) \cap A\neq\emptyset\right\} $). It is known that $α(G)-μ(G)\leq d\left( G\right) $ holds for every graph. A graph $G$ is unicyclic if it has a unique cycle and alm…
We show that the ratio of the number of near perfect matchings to the number of perfect matchings in $d$-regular strong expander (non-bipartite) graphs, with $2n$ vertices, is a polynomial in $n$, thus the Jerrum and Sinclair Markov chain [JS89] mixes in polynomial time and generates an (almost) uniformly random perfect matching. Furthermore, we prove that such graphs have at least $Ω(d)^n$ any perfect matchings, thus proving the Lovasz-Plummer conjecture [LP86] for this family of graphs.
Let $\G$ denote a bipartite distance-regular graph with vertex set $X$ and diameter $D \ge 3$. Fix $x \in X$ and let $L$ (resp. $R$) denote the corresponding lowering (resp. raising) matrix. We show that each $Q$-polynomial structure for $\G$ yields a certain linear dependency among $RL^2$, $LRL$, $L^2R$, $L$. Define a partial order $\le$ on $X$ as follows. For $y,z \in X$ let $y \le z$ whenever $\partial(x,y)+\partial(y,z)=\partial(x,z)$, where $\partial$ denotes path-length distance. We determine whether the above linear dependency gives this poset a uniform or strongly uniform structure. We…
We focus in this paper on edge ideals associated to bipartite graphs and give a combinatorial characterization of those having regularity 3. When the regularity is strictly bigger than 3, we determine the first step $i$ in the minimal graded free resolution where there exists a minimal generator of degree $>i+3$, show that at this step the highest degree of a minimal generator is $i+4$, and determine the value of the corresponding graded Betti number $β_{i,i+4}$ in terms of the combinatorics of the associated bipartite graph. The results can then be easily extended to the non-squarefree case t…
Let $G$ be a bipartite graph and $I=I(G)$ be its edge ideal. The aim of this note is to investigate different aspects of the Rees algebra $\mathcal{R}(I)$ of $I$. We compute its regularity and the universal Gröbner basis of its defining equations; interestingly, both of them are described in terms of the combinatorics of $G$. We apply these ideas to study the regularity of the powers of $I$. For any $s \ge \text{match}(G)+\lvert E(G) \rvert +1$ we prove that $\text{reg}(I^{s+1})=\text{reg}(I^s)+2$.
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