Regular Bipartite Graphs Research Articles

On the structure of periodic complex Horadam orbits

Numerous geometric patterns identified in nature, art or science can be generated from recurrent sequences, such as for example certain fractals or Fermat’s spiral. Fibonacci numbers in particular have been used to design search techniques, pseudo random-number generators and data structures. Complex Horadam sequences are a natural extension of Fibonacci sequence to complex numbers, involving four parameters (two initial values and two in the defining recursion), therefore successive sequence terms can be visualized in the complex plane. Here, a classification of the periodic orbits is proposed, based on divisibility relations between orders of generators (roots of the characteristic polynomial). Regular star polygons, bipartite graphs and multi-symmetric patterns can be recovered for selected parameter values. Some applications are also suggested.

Ferromagnetic Potts Model: Refined #BIS-hardness and Related Results

Recent results establish for 2-spin antiferromagnetic systems that the computational complexity of approximating the partition function on graphs of maximum degree D undergoes a phase transition that coincides with the uniqueness phase transition on the infinite D-regular tree. For the ferromagnetic Potts model we investigate whether analogous hardness results hold. Goldberg and Jerrum showed that approximating the partition function of the ferromagnetic Potts model is at least as hard as approximating the number of independent sets in bipartite graphs (#BIS-hardness). We improve this hardness result by establishing it for bipartite graphs of maximum degree D. We first present a detailed picture for the phase diagram for the infinite D-regular tree, giving a refined picture of its first-order phase transition and establishing the critical temperature for the coexistence of the disordered and ordered phases. We then prove for all temperatures below this critical temperature that it is #BIS-hard to approximate the partition function on bipartite graphs of maximum degree D. As a corollary, it is #BIS-hard to approximate the number of k-colorings on bipartite graphs of maximum degree D when k <= D/(2 ln D). The #BIS-hardness result for the ferromagnetic Potts model uses random bipartite regular graphs as a gadget in the reduction. The analysis of these random graphs relies on recent connections between the maxima of the expectation of their partition function, attractive fixpoints of the associated tree recursions, and induced matrix norms. We extend these connections to random regular graphs for all ferromagnetic models and establish the Bethe prediction for every ferromagnetic spin system on random regular graphs. We also prove for the ferromagnetic Potts model that the Swendsen-Wang algorithm is torpidly mixing on random D-regular graphs at the critical temperature for large q.

Reflexive cacti: A survey

A graph is called reexive if its second largest eigenvalue does not exceed 2. We survey the results on reexive cacti obtained in the last two decades. We also discuss various patterns of appearing of Smith graphs as subgraphs of reexive cacti. In the Appendix, we survey the recent results concerning reexive bipartite regular graphs.

Matchings in Benjamini–Schramm convergent graph sequences

We introduce the matching measure of a finite graph as the uniform distribution on the roots of the matching polynomial of the graph. We analyze the asymptotic behavior of the matching measure for graph sequences with bounded degree. A graph parameter is said to be estimable if it converges along every Benjamini–Schramm convergent sparse graph sequence. We prove that the normalized logarithm of the number of matchings is estimable. We also show that the analogous statement for perfect matchings already fails for d d –regular bipartite graphs for any fixed d ≥ 3 d\ge 3 . The latter result relies on analyzing the probability that a randomly chosen perfect matching contains a particular edge. However, for any sequence of d d –regular bipartite graphs converging to the d d –regular tree, we prove that the normalized logarithm of the number of perfect matchings converges. This applies to random d d –regular bipartite graphs. We show that the limit equals the exponent in Schrijver’s lower bound on the number of perfect matchings. Our analytic approach also yields a short proof for the Nguyen–Onak (also Elek–Lippner) theorem saying that the matching ratio is estimable. In fact, we prove the slightly stronger result that the independence ratio is estimable for claw-free graphs.

Antimagic Labeling of Regular Graphs

A graph $G=(V,E)$ is antimagic if there is a one-to-one correspondence $f: E \to \{1,2,\ldots, |E|\}$ such that for any two vertices $u,v$, $\sum_{e \in E(u)}f(e) \ne \sum_{e\in E(v)}f(e)$. It is known that bipartite regular graphs are antimagic and non-bipartite regular graphs of odd degree at least three are antimagic. Whether all non-bipartite regular graphs of even degree are antimagic remained an open problem. In this paper, we solve this problem and prove that all even degree regular graphs are antimagic. The paper was submitted to December 2014 by Journal of Graph Theory.

Interlacing families I: Bipartite Ramanujan graphs of all degrees

We prove that there exist infinite families of regular bipartite Ramanujan graphs of every degree bigger than 2. We do this by proving a variant of a conjecture of Bilu and Linial about the existence of good 2-lifts of every graph. We also establish the existence of infinite families of `irregular Ramanujan' graphs, whose eigenvalues are bounded by the spectral radius of their universal cover. Such families were conjectured to exist by Linial and others. In particular, we prove the existence of infinite families of (c,d)-biregular bipartite graphs with all non-trivial eigenvalues bounded by sqrt{c-1}+sqrt{d-1}, for all c, d \geq 3. Our proof exploits a new technique for demonstrating the existence of useful combinatorial objects that we call the method of interlacing polynomials'.

Semisymmetric graphs admitting primitive groups of degree 9p

Let Γ be a connected regular bipartite graph of order 18p, where p is a prime. Assume that Γ admits a group acting primitively on one of the bipartition subsets of Γ. Then, in this paper, it is shown that either Γ is arc-transitive, or Γ is isomorphic to one of 17 semisymmetric graphs which are constructed from primitive groups of degree 9p.

The cost of perfection for matchings in graphs

Perfect matchings and maximum weight matchings are two fundamental combinatorial structures. We consider the ratio between the maximum weight of a perfect matching and the maximum weight of a general matching. Motivated by the computer graphics application in triangle meshes, where we seek to convert a triangulation into a quadrangulation by merging pairs of adjacent triangles, we focus mainly on bridgeless cubic graphs.First, we characterize graphs that attain the extreme ratios. Second, we present a lower bound for all bridgeless cubic graphs. Third, we present upper bounds for subclasses of bridgeless cubic graphs, most of which are shown to be tight. Additionally, we present tight bounds for the class of regular bipartite graphs.

A positivity property of the dimer entropy of graphs

The entropy of a monomer–dimer system on an infinite regular bipartite lattice can be written as a mean-field part plus a series expansion in the dimer density. In a previous paper it has been conjectured that all coefficients of this series are positive. Analogously on a connected regular graph with v vertices, the “entropy” of the graph lnN(i)/v, where N(i) is the number of ways of setting down i dimers on the graph, can be written as a part depending only on the number of the dimer configurations over the completed graph plus a Newton series in the dimer density on the graph. In this paper, we investigate for which connected regular graphs all the coefficients of the Newton series are positive (for short, these graphs will be called positive). In the class of connected regular bipartite graphs, up to v=20, the only non positive graphs have vertices of degree 3. From v=14 to v=30, the frequency of the positivity violations in the 3-regular graphs decreases with increasing v. In the case of connected 4-regular bipartite graphs, the first violations occur in two out of the 2806490 graphs with v=22. We conjecture that for each degree r the frequency of the violations, in the class of the r-regular bipartite graphs, goes to zero as v tends to infinity.This graph-positivity property can be extended to non-regular or non-bipartite graphs.We have examined a large number of rectangular grids of size Nx×Ny both with open and periodic boundary conditions. We have observed positivity violations only for min(Nx,Ny)=3 or 4.

Upper bounds on the [formula omitted]-forcing number of a graph

Given a simple undirected graph G and a positive integer k, the k-forcing number of G, denoted Fk(G), is the minimum number of vertices that need to be initially colored so that all vertices eventually become colored during the discrete dynamical process described by the following rule. Starting from an initial set of colored vertices and stopping when all vertices are colored: if a colored vertex has at most k non-colored neighbors, then each of its non-colored neighbors becomes colored. When k=1, this is equivalent to the zero forcing number, usually denoted with Z(G), a recently introduced invariant that gives an upper bound on the maximum nullity of a graph. In this paper, we give several upper bounds on the k-forcing number. Notable among these, we show that if G is a graph with order n≥2 and maximum degree Δ≥k, then Fk(G)≤(Δ−k+1)nΔ−k+1+min{δ,k}. This simplifies to, for the zero forcing number case of k=1, Z(G)=F1(G)≤ΔnΔ+1. Moreover, when Δ≥2 and the graph is k-connected, we prove that Fk(G)≤(Δ−2)n+2Δ+k−2, which is an improvement when k≤2, and specializes to, for the zero forcing number case, Z(G)=F1(G)≤(Δ−2)n+2Δ−1. These results resolve a problem posed by Meyer about regular bipartite circulant graphs. Finally, we present a relationship between the k-forcing number and the connected k-domination number. As a corollary, we find that the sum of the zero forcing number and connected domination number is at most the order for connected graphs.

Note on the Subgraph Component Polynomial

Tittmann, Averbouch and Makowsky [The enumeration of vertex induced subgraphs with respect to the number of components, European J. Combin. 32 (2011) 954-974] introduced the subgraph component polynomial $Q(G;x,y)$ of a graph $G$, which counts the number of connected components in vertex induced subgraphs. This polynomial encodes a large amount of combinatorial information about the underlying graph, such as the order, the size, and the independence number. We show that several other graph invariants, such as the connectivity and the number of cycles of length four in a regular bipartite graph are also determined by the subgraph component polynomial. Then, we prove that several well-known families of graphs are determined by the polynomial $Q(G;x,y).$ Moreover, we study the distinguishing power and find simple graphs which are not distinguished by the subgraph component polynomial but distinguished by the characteristic polynomial, the matching polynomial and the Tutte polynomial. These are partial answers to three open problems proposed by Tittmann et al.

On strongly [formula omitted]-connected graphs

An orientation of a graph G is a mod(2s+1)-orientation if under this orientation, the net out-degree at every vertex is congruent to zero mod(2s+1). If for any function b:V(G)→Z2s+1 satisfying ∑v∈V(G)b(v)≡0(mod2s+1), G always has an orientation D such that the net out-degree at every vertex v is congruent to b(v)mod(2s+1), then G is strongly Z2s+1-connected. In this paper, we prove that a connected graph has a mod(2s+1)-orientation if and only if it is a contraction of a (2s+1)-regular bipartite graph. We also proved that every (4s−1)-edge-connected series–parallel graph is strongly Z2s+1-connected, and every simple 4p-connected chordal graph is strongly Z2s+1-connected.

Hardness and approximation of minimum maximal matchings

In this paper, we consider the minimum maximal matching problem in some classes of graphs such as regular graphs. We show that the minimum maximal matching problem is NP-hard even in regular bipartite graphs, and a polynomial time exact algorithm is given for almost complete regular bipartite graphs. From the approximation point of view, it is well known that any maximal matching guarantees the approximation ratio of 2 but surprisingly very few improvements have been obtained. In this paper we give improved approximation ratios for several classes of graphs. For example any algorithm is shown to guarantee an approximation ratio of (2-o(1)) in graphs with high average degree. We also propose an algorithm guaranteeing for any graph of maximum degree Δ an approximation ratio of (2−1/Δ), which slightly improves the best known results. In addition, we analyse a natural linear-time greedy algorithm guaranteeing a ratio of (2−23/18k) in k-regular graphs admitting a perfect matching.

Interval incidence coloring of bipartite graphs

In this paper11The extended abstract of this paper appeared on PPAM 2009, LNCS 6067 (2010) 11–20. we study the problem of interval incidence coloring of bipartite graphs. We show the upper bound for interval incidence coloring number (χii) for bipartite graphs χii≤2Δ, and we prove that χii=2Δ holds for regular bipartite graphs. We solve this problem for subcubic bipartite graphs, i.e. we fully characterize the subcubic graphs that admit 4, 5 or 6 coloring, and we construct a linear time exact algorithm for subcubic bipartite graphs. We also study the problem for bipartite graphs with Δ=4 and we show that 5-coloring is easy and 6-coloring is hard (NP-complete). Moreover, we construct an O(nΔ3.5logΔ) time optimal algorithm for trees.

Can Colour-Blind Distinguish Colour Palettes?

Let $c:E(G)\rightarrow [k]$ be a colouring, not necessarily proper, of edges of a graph $G$. For a vertex $v\in V$, let $\overline{c}(v)=(a_1,\ldots,a_k)$, where $ a_i =|\{u:uv\in E(G),\;c(uv)=i\}|$, for $i\in [k].$ If we re-order the sequence $\overline{c}(v)$ non-decreasingly, we obtain a sequence $c^*(v)=(d_1,\ldots,d_k)$, called a palette of a vertex $v$. This can be viewed as the most comprehensive information about colours incident with $v$ which can be delivered by a person who is unable to name colours but distinguishes one from another. The smallest $k$ such that $c^*$ is a proper colouring of vertices of $G$ is called the colour-blind index of a graph $G$, and is denoted by dal$(G)$. We conjecture that there is a constant $K$ such that dal$(G)\leq K$ for every graph $G$ for which the parameter is well defined. As our main result we prove that $K\leq 6$ for regular graphs of sufficiently large degree, and for irregular graphs with $\delta (G)$ and $\Delta(G)$ satisfying certain conditions. The proofs are based on the Lopsided Lovász Local Lemma. We also show that $K=3$ for all regular bipartite graphs, and for complete graphs of order $n\geq 8$.

Reflexive bipartite regular graphs

A graph is called reflexive if its second largest eigenvalue does not exceed 2. In this paper, we determine all reflexive bipartite regular graphs. Any bipartite regular graph of degree at most 2 is reflexive as well as its bipartite complement. Apart from them, there is a finite number of resulting graphs.

Extensions to 2-Factors in Bipartite Graphs

A graph is $d$-bounded if its maximum degree is at most $d$. We apply the Ore-Ryser Theorem on $f$-factors in bipartite graphs to obtain conditions for the extension of a $2$-bounded subgraph to a $2$-factor in a bipartite graph. As consequences, we prove that every matching in the $5$-dimensional hypercube extends to a $2$-factor, and we obtain conditions for this property in general regular bipartite graphs. For example, to show that every matching in a regular $n$-vertex bipartite graph extends to a $2$-factor, it suffices to show that all matchings with fewer than $n/3$ edges extend to $2$-factors.

Efficient crowdsourcing for multi-class labeling

Crowdsourcing systems like Amazon's Mechanical Turk have emerged as an effective large-scale human-powered platform for performing tasks in domains such as image classification, data entry, recommendation, and proofreading. Since workers are low-paid (a few cents per task) and tasks performed are monotonous, the answers obtained are noisy and hence unreliable. To obtain reliable estimates, it is essential to utilize appropriate inference algorithms (e.g. Majority voting) coupled with structured redundancy through task assignment. Our goal is to obtain the best possible trade-off between reliability and redundancy. In this paper, we consider a general probabilistic model for noisy observations for crowd-sourcing systems and pose the problem of minimizing the total price (i.e. redundancy) that must be paid to achieve a target overall reliability. Concretely, we show that it is possible to obtain an answer to each task correctly with probability 1-ε as long as the redundancy per task is O((K/q) log (K/ε)), where each task can have any of the $K$ distinct answers equally likely, q is the crowd-quality parameter that is defined through a probabilistic model. Further, effectively this is the best possible redundancy-accuracy trade-off any system design can achieve. Such a single-parameter crisp characterization of the (order-)optimal trade-off between redundancy and reliability has various useful operational consequences. Further, we analyze the robustness of our approach in the presence of adversarial workers and provide a bound on their influence on the redundancy-accuracy trade-off. Unlike recent prior work [GKM11, KOS11, KOS11], our result applies to non-binary (i.e. K&gt;2) tasks. In effect, we utilize algorithms for binary tasks (with inhomogeneous error model unlike that in [GKM11, KOS11, KOS11]) as key subroutine to obtain answers for K-ary tasks. Technically, the algorithm is based on low-rank approximation of weighted adjacency matrix for a random regular bipartite graph, weighted according to the answers provided by the workers.

Algebraic characterizations of regularity properties in bipartite graphs

Regular and distance-regular characterizations of general graphs are well-known. In particular, the spectral excess theorem states that a connected graph Γ 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 of the mean degrees in every partite set and the Hoffman polynomial. Moreover, it is shown that the conditions for having distance-regularity in such graphs can be relaxed when compared with general graphs. Finally, a new version of the spectral excess theorem for bipartite graphs is presented.

Regular graphs with girth at least 5 and small second largest eigenvalue

In this paper we present an upper bound on the degree of a regular graph with girth at least 5 in terms of its second largest eigenvalue (usually denoted by λ2). Next, we consider bipartite r-regular graphs with girth at least 6 whose second largest eigenvalue satisfies λ2⩽r and prove that such graphs are the incidence graphs of two-class partially balanced incomplete block designs and also balanced incomplete block designs. Upper bound on the order of such graphs is also given. We also consider the relations between the incidence graphs of two-class partially balanced incomplete block designs and distance-regular graphs. In particular, we determine all regular reflexive graphs of degree 3 with girth at least 5, all regular graphs with girth at least 5 satisfying λ2⩽3 and all bipartite regular reflexive graphs with girth at least 6.