On a conjecture of Erdős and Simonovits: Even cycles

Let $\mc{F}$ be a family of graphs. A graph is {\em $\mc{F}$-free} if it contains no copy of a graph in $\mc{F}$ as a subgraph. A cornerstone of extremal graph theory is the study of the {\em Tur\'an number} $ex(n,\mc{F})$, the maximum number of edges in an $\mc{F}$-free graph on $n$ vertices. Define the {\em Zarankiewicz number} $z(n,\mc{F})$ to be the maximum number of edges in an $\mc{F}$-free {\em bipartite} graph on $n$ vertices. Let $C_k$ denote a cycle of length $k$, and let $\mc{C}_k$ denote the set of cycles $C_{\ell}$, where $3 \le \ell \leq k$ and $\ell$ and $k$ have the same parity. Erd\H{o}s and Simonovits conjectured that for any family $\mc{F}$ consisting of bipartite graphs there exists an odd integer $k$ such that $ex(n,\mc{F} \cup \mc{C}_k) \sim z(n,\mc{F})$. They proved this when $\mc{F}={C_4}$ by showing that $ex(n,\{C_4,C_5\}) \sim z(n,C_4)$. In this paper, we extend this result by showing that if $\ell \in \{2,3,5\}$ and $k > 2\ell$ is odd, then ${ex(n,\mc{C}_{2\ell} \cup {C_k}) \sim z(n,\mc{C}_{2\ell})$. Furthermore, if $k > 2\ell + 2$ is odd, then for infinitely many $n$ we show that the extremal $\mc{C}_{2\ell} \cup \{C_k\}$-free graphs are bipartite incidence graphs of generalized polygons. We observe that this exact result does not hold for any odd $k < 2\ell$, and furthermore the asymptotic result does not hold when $(\ell,k)$ is $(3,3)$, $(5,3)$ or $(5,5)$. Our proofs make use of pseudorandomness properties of nearly extremal graphs that are of independent interest.

Turán numbers of bipartite graphs plus an odd cycle

Constructing extremal triangle-free graphs using integer programming

On edges not in monochromatic copies of a fixed bipartite graph

We show that it is NP-complete to determine the chromatic index of an arbitrary graph. The problem remains NP-complete even for cubic graphs.

The Erdős–Simonovits stability theorem is one of the most widely used theorems in extremal graph theory. We obtain an Erdős–Simonovits type stability theorem in multi-partite graphs. Different from the Erdős–Simonovits stability theorem, our stability theorem in multi-partite graphs says that if the number of edges of an $H$ -free graph $G$ is close to the extremal graphs for $H$ , then $G$ has a well-defined structure but may be far away from the extremal graphs for $H$ . As applications, we strengthen a theorem of Bollobás, Erdős, and Straus and solve a conjecture in a stronger form posed by Han and Zhao concerning the maximum number of edges in multi-partite graphs which does not contain vertex-disjoint copies of a clique.

Maximum biclique search, which finds the biclique with the maximum number of edges in a bipartite graph, is a fundamental problem with a wide spectrum of applications in different domains, such as E-Commerce, social analysis, web services, and bioinformatics. Unfortunately, due to the difficulty of the problem in graph theory, no practical solution has been proposed to solve the issue in large-scale real-world datasets. Existing techniques for maximum clique search on a general graph cannot be applied because the search objective of maximum biclique search is two-dimensional, i.e., we have to consider the size of both parts of the biclique simultaneously. In this paper, we divide the problem into several subproblems each of which is specified using two parameters. These subproblems are derived in a progressive manner, and in each subproblem, we can restrict the search in a very small part of the original bipartite graph. We prove that a logarithmic number of subproblems is enough to guarantee the algorithm correctness. To minimize the computational cost, we show how to reduce significantly the bipartite graph size for each subproblem while preserving the maximum biclique satisfying certain constraints by exploring the properties of one-hop and two-hop neighbors for each vertex. Furthermore, we study the diversified top-k biclique search problem which aims to find k maximal bicliques that cover the most edges in total. The basic idea is to repeatedly find the maximum biclique in the bipartite graph and remove it from the bipartite graph k times. We design an efficient algorithm that considers to share the computation cost among the k results, based on the idea of deriving the same subproblems of different results. We further propose two optimizations to accelerate the computation by pruning the search space with size constraint and refining the candidates in a lazy manner. We use several real datasets from various application domains, one of which contains over 300 million vertices and 1.3 billion edges, to demonstrate the high efficiency and scalability of our proposed solution. It is reported that 50% improvement on recall can be achieved after applying our method in Alibaba Group to identify the fraudulent transactions in their e-commerce networks. This further demonstrates the usefulness of our techniques in practice.

Extremal <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msub><mml:mi>K</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>s</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msub></mml:math>-free bipartite graphs

The bipartite Turán number and spectral extremum for linear forests

Let F be a family of graphs. A graph is called F-free if it does not contain any member of F as a subgraph. The Turán number of F is the maximum number of edges in an n-vertex F-free graph and is denoted by ex(n,F). The same maximum under the additional condition that the graphs are connected is exconn(n,F). Let Pk be the path on k vertices, Km be the clique on m vertices. We determine ex(n,{Pk,Km}) if k>2m−1 and exconn(n,{Pk,Km}) if k>m for sufficiently large n.

Constant factor approximation algorithms for the densest k-subgraph problem on proper interval graphs and bipartite permutation graphs

Induced matchings in bipartite graphs

For a given graph G, a maximum internal spanning tree of G is a spanning tree of G with maximum number of internal vertices. The Maximum Internal Spanning Tree (MIST) problem is to find a maximum internal spanning tree of the given graph. The MIST problem is a generalization of the Hamiltonian path problem. Since the Hamiltonian path problem is NP-hard, even for bipartite and chordal graphs, two important subclasses of graphs, the MIST problem also remains NP-hard for these graph classes. In this paper, we propose linear-time algorithms to compute a maximum internal spanning tree of cographs, block graphs, cactus graphs, chain graphs and bipartite permutation graphs. The optimal path cover problem, which asks to find a path cover of the given graph with maximum number of edges, is also a well studied problem. In this paper, we also study the relationship between the number of internal vertices in maximum internal spanning tree and number of edges in optimal path cover for the special graph classes mentioned above.

On extremal bipartite graphs with high girth

Let $F$ be a graph. A graph $G$ is $F$-free if it does not contain $F$ as a subgraph. The Turán number of $F$, written $\textrm{ex}(n,F)$, is the maximum number of edges in an $F$-free graph with $n$ vertices. The determination of Turán numbers of bipartite graphs is a challenging and widely investigated problem. In this paper we introduce an ordered version of the Turán problem for bipartite graphs. Let $G$ be a graph with $V(G) = \{1, 2, \dots , n \}$ and view the vertices of $G$ as being ordered in the natural way. A zig-zag $K_{s,t}$, denoted $Z_{s,t}$, is a complete bipartite graph $K_{s,t}$ whose parts $A = \{n_1 &lt; n_2 &lt; \dots &lt; n_s \}$ and $B = \{m_1 &lt; m_2 &lt; \dots &lt; m_t \}$ satisfy the condition $n_s &lt; m_1$. A zig-zag $C_{2k}$ is an even cycle $C_{2k}$ whose vertices in one part precede all of those in the other part. Write $\mathcal{Z}_{2k}$ for the family of zig-zag $2k$-cycles. We investigate the Turán numbers $\textrm{ex}(n,Z_{s,t})$ and $\textrm{ex}(n,\mathcal{Z}_{2k})$. In particular we show $\textrm{ex}(n, Z_{2,2}) \leq \frac{2}{3}n^{3/2} + O(n^{5/4})$. For infinitely many $n$ we construct a $Z_{2,2}$-free $n$-vertex graph with more than $(n - \sqrt{n} - 1) + \textrm{ex} (n,K_{2,2})$ edges.