Generalized Planar Turán Numbers

In a generalized Turán problem, we are given graphs $H$ and $F$ and seek to maximize the number of copies of $H$ in an $n$-vertex graph not containing $F$ as a subgraph. We consider generalized Turán problems where the host graph is planar. In particular, we obtain the order of magnitude of the maximum number of copies of a fixed tree in a planar graph containing no even cycle of length at most $2\ell$, for all $\ell$, $\ell \geqslant 1$. We also determine the order of magnitude of the maximum number of cycles of a given length in a planar $C_4$-free graph. An exact result is given for the maximum number of $5$-cycles in a $C_4$-free planar graph. Multiple conjectures are also introduced.

Faster algorithms for counting subgraphs in sparse graphs

Given a k-node pattern graph H and an n-node host graph G, the subgraph counting problem asks to compute the number of copies of H in G. In this work we address the following question: can we count the copies of H faster if G is sparse? We answer in the affirmative by introducing a novel tree-like decomposition for directed acyclic graphs, inspired by the classic tree decomposition for undirected graphs. This decomposition gives a dynamic program for counting the homomorphisms of H in G by exploiting the degeneracy of G, which allows us to beat the state-of-the-art subgraph counting algorithms when G is sparse enough. For example, we can count the induced copies of any k-node pattern H in time $$2^{O(k^2)} O(n^{0.25k + 2} \log n)$$ 2 O ( k 2 ) O ( n 0.25 k + 2 log n ) if G has bounded degeneracy, and in time $$2^{O(k^2)} O(n^{0.625k + 2} \log n)$$ 2 O ( k 2 ) O ( n 0.625 k + 2 log n ) if G has bounded average degree. These bounds are instantiations of a more general result, parameterized by the degeneracy of G and the structure of H, which generalizes classic bounds on counting cliques and complete bipartite graphs. We also give lower bounds based on the Exponential Time Hypothesis, showing that our results are actually a characterization of the complexity of subgraph counting in bounded-degeneracy graphs.

The maximum number of triangles in a graph of given maximum degree

Maximizing or minimizing the number of copies of a fixed graph in a large host graph is one of the most classical topics in extremal graph theory. Indeed, one of the most famous problems in extremal graph theory, the Erdős-Rademacher problem, which can be traced back to the 1940s, asks to determine the minimum number of triangles in a graph with a given number of vertices and edges. It was conjectured that the mnimum is attained by complete multipartite graphs with all parts but one of the same size whilst the remaining part may be smaller. The problem was widely open in the regime of four or more parts until Razborov resolved the problem asymptotically in 2008 as one of the first applications of his newly developed flag algebra method. This catalyzed a line of research on the structure of extremal graphs and extensions. In particular, Reiher asymptotically solved in 2016 the conjecture of Lovász and Simonovits from the 1970s that the same graphs are also minimizers for cliques of arbitrary size. This paper deals with a problem concerning the opposite direction: _What is the maximum number of triangles in a graph with a given number $n$ of vertices and a given maximum degree $D$?_ Gan, Loh and Sudakov in 2015 conjectured that the graph maximizing the number of triangles is always a union of disjoint cliques of size $D+1$ and another clique that may be smaller, and showed that if such a graph maximizes the number of triangles, it also maximizes the number of cliques of any size $r\ge 4$. The author presents a remarkably simple and elegant argument that proves the conjecture exactly for all $n$ and $D$.

Rainbow Matchings of size $m$ in Graphs with Total Color Degree at least $2mn$

The existence of a rainbow matching given a minimum color degree, proper coloring, or triangle-free host graph has been studied extensively. This paper generalizes these problems to edge colored graphs with given total color degree. In particular, we find that if a graph $G$ has total color degree $2mn$ and satisfies some other properties, then $G$ contains a matching of size $m$. These other properties include $G$ being triangle-free, $C_4$-free, properly colored, or large enough.

Optimal Graphs in the Enhanced Mesh Networks

The degree diameter problem explores the biggest graph (in terms of number of nodes) subject to some restrictions on the valency and the diameter of the graph. The restriction on the valency of the graph does not impose any condition on the number of edges (apart from taking the graph simple), so the resulting graph may be thought of as being embedded in the complete graph. In a generality of the said problem, the graph is taken to be embedded in any connected host graph. In this article, host graph is considered as the enhanced mesh network constructed from the grid network. This article provides some exact values for the said problem and also gives some bounds for the optimal graphs.

A rainbow blow-up lemma for almost optimally bounded edge-colourings

A subgraph of an edge-coloured graph is called rainbow if all its edges have different colours. We prove a rainbow version of the blow-up lemma of Komlós, Sárközy, and Szemerédi that applies to almost optimally bounded colourings. A corollary of this is that there exists a rainbow copy of any bounded-degree spanning subgraph H in a quasirandom host graph G, assuming that the edge-colouring of G fulfills a boundedness condition that is asymptotically best possible. This has many applications beyond rainbow colourings: for example, to graph decompositions, orthogonal double covers, and graph labellings.

Fault-Tolerant Path-Embedding of Twisted Hypercube-Like Networks (THLNs)

It is known widely that an interconnection network can be denoted by a graph G = ( V , E ) , where V denotes the vertex set and E denotes the edge set. Investigating structures of G is necessary to design a suitable topological structure of interconnection network. One of the critical issues in evaluating an interconnection network is graph embedding, which concerns whether a host graph contains a guest graph as its subgraph. Linear arrays (i.e., paths) and rings (i.e., cycles) are two ordinary guest graphs (or basic networks) for parallel and distributed computation. In the process of large-scale interconnection network operation, it is inevitable that various errors may occur at nodes and edges. It is significant to find an embedding of a guest graph into a host graph where all faulty nodes and edges have been removed. This is named as fault-tolerant embedding. The twisted hypercube-like networks ( T H L N s ) contain several important hypercube variants. This paper is concerned with the fault-tolerant path-embedding of n-dimensional (n-D) T H L N s . Let G n be an n-D T H L N and F be a subset of V ( G n ) ∪ E ( G n ) with | F | ≤ n - 2 . We show that for two different arbitrary correct vertices u and v, there is a faultless path P u v of every length l with 2 n - 1 - 1 ≤ l ≤ 2 n - f v - 1 - α , where α = 0 if vertices u and v form a normal vertex-pair and α = 1 if vertices u and v form a weak vertex-pair in G n - F ( n ≥ 5 ).

On the Chromatic Number of Matching Kneser Graphs

In an earlier paper, the present authors (2015) introduced the altermatic number of graphs and used Tucker’s lemma, an equivalent combinatorial version of the Borsuk–Ulam theorem, to prove that the altermatic number is a lower bound for chromatic number. A matching Kneser graph is a graph whose vertex set consists of all matchings of a specified size in a host graph and two vertices are adjacent if their corresponding matchings are edge-disjoint. Some well-known families of graphs such as Kneser graphs, Schrijver graphs and permutation graphs can be represented by matching Kneser graphs. In this paper, unifying and generalizing some earlier works by Lovász (1978) and Schrijver (1978), we determine the chromatic number of a large family of matching Kneser graphs by specifying their altermatic number. In particular, we determine the chromatic number of these matching Kneser graphs in terms of the generalized Turán number of matchings.

MaxDDBS Problem on Beneš Network

Maximum degree-diameter bounded subgraph problem is a problem of constructing the largest possible subgraph of given degree and diameter in a graph. This problem can be considered as a degree-diameter problem restricted to certain host graphs. The MaxDDBS problem with Beneš network as the host graph is discussed in this paper. Beneš network contains a back-to-back buttery network. Even though both networks have maximum degree 4, the structure of their maximum subgraphs are different. We give the constructive lower bound of the largest subgraph of Beneš network of various maximum degrees.