On the hardness of recognizing triangular line graphs

Given a graph G, its triangular line graph is the graph T(G) with vertex set consisting of the edges of G and adjacencies between edges that are incident in G as well as being within a common triangle. Graphs with a representation as the triangular line graph of some graph G are triangular line graphs, which have been studied under many names including anti-Gallai graphs, 2-in-3 graphs, and link graphs. While closely related to line graphs, triangular line graphs have been difficult to understand and characterize. Van Bang Le asked if recognizing triangular line graphs has an efficient algorithm or is computationally complex. We answer this question by proving that the complexity of recognizing triangular line graphs is NP-complete via a reduction from 3-SAT.Comment: 18 pages, 8 figures, 4 table

Eulerian and Hamiltonian properties of Gallai and anti-Gallai middle graphs

The Gallai middle graph ΓM(G) of a graph G = (V, E) is the graph whose vertex set is V ∪ E and two edges ei, ej ∈ E are adjacent in ΓM(G), if they are adjacent edges of G and do not lie on a same triangle in G, or if ei = uv ∈ E then ei is adjacent to u and v in ΓM(G). The anti-Gallai middle graph ∆M(G) of a graph G = (V, E) is the graph whose vertex set is V ∪ E and two edges ei, ej ∈ E are adjacent in ∆M(G) if they are adjacent in G and lie on a same triangle in G, or if ei = uv ∈ E then ei is adjacent to u and v in ∆M(G). In this paper, we investigate Eulerian and Hamiltonian properties of Gallai and anti-Gallai middle graphs.Publisher's Versio

Gallai-Edmonds percolation of topologically protected collective Majorana excitations

Majorana networks, whose vertices represent localized Majorana modes and edges correspond to bilinear mixing amplitudes between them, provide a unified framework for describing the low energy physics of several interesting systems. Such networks are known to exhibit topologically protected collective Majorana modes if the combinatorial problem of maximum matchings (maximally-packed dimer covers) of the underlying graph has unmatched vertices (monomers), as is typically the case if the network is disordered. These collective Majorana modes live in ``${\mathcal R}$-type regions'' of the disordered graph, which host the unmatched vertices (monomers) in any maximum matching and can be identified using the graph theoretical Gallai-Edmonds decomposition. Here, we focus on vacancy disorder (site dilution) in general (nonbipartite) two dimensional lattices such as the triangular and Shastry-Sutherland lattices, and study the random geometry of such ${\mathcal R}$-type regions and their complements, i.e., ``${\mathcal P}$-type regions'' from which monomers are excluded in any maximum matching of the lattice. These ${\mathcal R}$-type and ${\mathcal P}$-type regions are found to display a sharply-defined {\em Gallai-Edmonds percolation} transition at a critical vacancy density $n_v^{\rm crit}$ that lies well within the geometrically percolated phase of the underlying disordered lattice. For $n_v<n_v^{\rm crit}$, ${\mathcal R}$-type regions percolate but there is a striking lack of self-averaging even in the thermodynamic limit, with the ensemble average being macroscopically different from the properties of individual samples: Each sample has exactly one infinite cluster, which is of type ${\mathcal R}$ ($\mathcal P$) in a weakly $n_v$-dependent fraction $f <1$ ($1-f < 1$) of the samples

Algorithmic Graph Theory

The main focus of this workshop was on mathematical techniques needed for the development of efficient solutions and algorithms for computationally difficult graph problems. The techniques studied at the workshhop included: the probabilistic method and randomized algorithms, approximation and optimization, structured families of graphs and approximation algorithms for large problems. The workshop Algorithmic Graph Theory was attended by 46 participants, many of them being young researchers. In 15 survey talks an overview of recent developments in Algorithmic Graph Theory was given. These talks were supplemented by 10 shorter talks and by two special sessions

Global hypercontractivity and its applications

The hypercontractive inequality on the discrete cube plays a crucial role in many fundamental results in the Analysis of Boolean functions, such as the KKL theorem, Friedgut's junta theorem and the invariance principle. In these results the cube is equipped with the uniform measure, but it is desirable, particularly for applications to the theory of sharp thresholds, to also obtain such results for general $p$-biased measures. However, simple examples show that when $p = o(1)$, there is no hypercontractive inequality that is strong enough. In this paper, we establish an effective hypercontractive inequality for general $p$ that applies to `global functions', i.e. functions that are not significantly affected by a restriction of a small set of coordinates. This class of functions appears naturally, e.g. in Bourgain's sharp threshold theorem, which states that such functions exhibit a sharp threshold. We demonstrate the power of our tool by strengthening Bourgain's theorem, thereby making progress on a conjecture of Kahn and Kalai and by establishing a $p$-biased analog of the invariance principle. Our results have significant applications in Extremal Combinatorics. Here we obtain new results on the Tur\'an number of any bounded degree uniform hypergraph obtained as the expansion of a hypergraph of bounded uniformity. These are asymptotically sharp over an essentially optimal regime for both the uniformity and the number of edges and solve a number of open problems in the area. In particular, we give general conditions under which the crosscut parameter asymptotically determines the Tur\'an number, answering a question of Mubayi and Verstra\"ete. We also apply the Junta Method to refine our asymptotic results and obtain several exact results, including proofs of the Huang--Loh--Sudakov conjecture on cross matchings and the F\"uredi--Jiang--Seiver conjecture on path expansions.Comment: Subsumes arXiv:1906.0556

New graph invariants based on pp-Laplacian eigenvalues

We present monotonicity inequalities for certain functions involving eigenvalues of $p$-Laplacians on signed graphs with respect to $p$. Inspired by such monotonicity, we propose new spectrum-based graph invariants, called (variational) cut-off adjacency eigenvalues, that are relevant to certain eigenvector-dependent nonlinear eigenvalue problem. Using these invariants, we obtain new lower bounds for the $p$-Laplacian variational eigenvalues, essentially giving the state-of-the-art spectral asymptotics for these eigenvalues. Moreover, based on such invariants, we establish two inertia bounds regarding the cardinalities of a maximum independent set and a minimum edge cover, respectively. The first inertia bound enhances the classical Cvetkovi\'c bound, and the second one implies that the $k$-th $p$-Laplacian variational eigenvalue is of the order $2^p$ as $p$ tends to infinity whenever $k$ is larger than the cardinality of a minimum edge cover of the underlying graph. We further discover an interesting connection between graph $p$-Laplacian eigenvalues and tensor eigenvalues and discuss applications of our invariants to spectral problems of tensors.Comment: 30 page