Unsolved Problems in Spectral Graph Theory

Spectral graph theory is a captivating area of graph theory that employs the eigenvalues and eigenvectors of matrices associated with graphs to study them. In this paper, we present a collection of $20$ topics in spectral graph theory, covering a range of open problems and conjectures. Our focus is primarily on the adjacency matrix of graphs, and for each topic, we provide a brief historical overview.Comment: v3, 30 pages, 1 figure, include comments from Clive Elphick, Xiaofeng Gu, William Linz, and Dragan Stevanovi\'c, respectively. Thanks! This paper will be published in Operations Research Transaction

Coloring, List Coloring, and Painting Squares of Graphs (and other related problems)

We survey work on coloring, list coloring, and painting squares of graphs; in particular, we consider strong edge-coloring. We focus primarily on planar graphs and other sparse classes of graphs.Comment: 32 pages, 13 figures and tables, plus 195-entry bibliography, comments are welcome, published as a Dynamic Survey in Electronic Journal of Combinatoric

Strong edge colorings of graphs and the covers of Kneser graphs

A proper edge coloring of a graph is strong if it creates no bichromatic path of length three. It is well known that for a strong edge coloring of a $k$-regular graph at least $2k-1$ colors are needed. We show that a $k$-regular graph admits a strong edge coloring with $2k-1$ colors if and only if it covers the Kneser graph $K(2k-1,k-1)$. In particular, a cubic graph is strongly $5$-edge-colorable whenever it covers the Petersen graph. One of the implications of this result is that a conjecture about strong edge colorings of subcubic graphs proposed by Faudree et al. [Ars Combin. 29 B (1990), 205--211] is false

New Graph Decompositions and Combinatorial Boolean Matrix Multiplication Algorithms

We revisit the fundamental Boolean Matrix Multiplication (BMM) problem. With the invention of algebraic fast matrix multiplication over 50 years ago, it also became known that BMM can be solved in truly subcubic $O(n^\omega)$ time, where $\omega<3$; much work has gone into bringing $\omega$ closer to $2$. Since then, a parallel line of work has sought comparably fast combinatorial algorithms but with limited success. The naive $O(n^3)$-time algorithm was initially improved by a $\log^2{n}$ factor [Arlazarov et al.; RAS'70], then by $\log^{2.25}{n}$ [Bansal and Williams; FOCS'09], then by $\log^3{n}$ [Chan; SODA'15], and finally by $\log^4{n}$ [Yu; ICALP'15]. We design a combinatorial algorithm for BMM running in time $n^3 / 2^{\Omega(\sqrt[7]{\log n})}$ -- a speed-up over cubic time that is stronger than any poly-log factor. This comes tantalizingly close to refuting the conjecture from the 90s that truly subcubic combinatorial algorithms for BMM are impossible. This popular conjecture is the basis for dozens of fine-grained hardness results. Our main technical contribution is a new regularity decomposition theorem for Boolean matrices (or equivalently, bipartite graphs) under a notion of regularity that was recently introduced and analyzed analytically in the context of communication complexity [Kelley, Lovett, Meka; arXiv'23], and is related to a similar notion from the recent work on $3$-term arithmetic progression free sets [Kelley, Meka; FOCS'23]

Being Even Slightly Shallow Makes Life Hard

We study the computational complexity of identifying dense substructures, namely r/2-shallow topological minors and r-subdivisions. Of particular interest is the case r = 1, when these substructures correspond to very localized relaxations of subgraphs. Since Densest Subgraph can be solved in polynomial time, we ask whether these slight relaxations also admit efficient algorithms. In the following, we provide a negative answer: Dense r/2-Shallow Topological Minor and Dense r-Subdivsion are already NP-hard for r = 1 in very sparse graphs. Further, they do not admit algorithms with running time 2^(o(tw^2)) n^O(1) when parameterized by the treewidth of the input graph for r > 2 unless ETH fails

On cycles and independence in graphs

Das erste Fachkapitel ist der Berechnung von Kreispackungszahlen, d.h. der maximalen Größe kanten- bzw. eckendisjunkter Kreispackungen, gewidmet. Da diese Probleme bekanntermaßen sogar für subkubische Graphen schwer sind, behandelt der erste Abschnitt die Komplexität des Packens von Kreisen einer festen Länge l in Graphen mit Maximalgrad Delta. Dieses für l=3 von Caprara und Rizzi gelöste Problem wird hier auf alle größeren Kreislängen l verallgemeinert. Der zweite Abschnitt beschreibt die Struktur von Graphen, für die die Kreispackungszahlen einen vorgegebenen Abstand zur zyklomatischen Zahl haben. Die 2-zusammenhängenden Graphen mit dieser Eigenschaft können jeweils durch Anwendung einer einfachen Erweiterungsregel auf eine endliche Menge von Graphen erzeugt werden. Aus diesem Strukturergebnis wird ein fpt-Algorithmus abgeleitet. Das zweite Fachkapitel handelt von der Größenordnung der minimalen Anzahl von Kreislängen in einem Hamiltongraph mit q Sehnen. Eine Familie von Beispielen zeigt, dass diese Unterschranke höchstens die Wurzel von q+1 ist. Dem Hauptsatz dieses Kapitels zufolge ist die Zahl der Kreislängen eines beliebigen Hamiltongraphen mit q Sehnen mindestens die Wurzel von 4/7*q. Der Beweis beruht auf einem Lemma von Faudree et al., demzufolge der Graph, der aus einem Weg mit Endecken x und y und q gleichlangen Sehnen besteht, x-y-Wege von mindestens q/3 verschiedenen Längen enthält. Der erste Abschnitt enthält eine Korrektur des ursprünglich fehlerhaften Beweises und zusätzliche Schranken. Der zweite Abschnitt leitet daraus die Unterschranke für die Anzahl der Kreislängen ab. Das letzte Fachkapitel behandelt Unterschranken für den Unabhängigkeitsquotienten, d.h. den Quotienten aus Unabhängigkeitszahl und Ordnung eines Graphen, für Graphen gegebener Dichte. In der Einleitung werden bestmögliche Schranken für die Klasse aller Graphen sowie für große zusammenhängende Graphen aus bekannten Ergebnissen abgeleitet. Danach wird die Untersuchung auf durch das Verbot kleiner ungerader Kreise eingeschränkte Graphenklassen ausgeweitet. Das Hauptergebnis des ersten Abschnitts ist eine Verallgemeinerung eines Ergebnisses von Heckman und Thomas, das die bestmögliche Schranke für zusammenhängende dreiecksfreie Graphen mit Durchschnittsgrad bis zu 10/3 impliziert und die extremalen Graphen charakterisiert. Der Rest der ersten beiden Abschnitte enthält Vermutungen ähnlichen Typs für zusammenhängende dreiecksfreie Graphen mit Durchschnittsgrad im Intervall [10/3, 54/13] und für zusammenhängende Graphen mit ungerader Taillenweite 7 mit Durchschnittsgrad bis zu 14/5. Der letzte Abschnitt enthält analoge Beobachtungen zum Bipartitionsquotienten. Die Arbeit schließt mit Vermutungen zu Unterschranken und die zugehörigen Klassen extremaler Graphen für den Bipartitionsquotienten.This thesis discusses several problems related to cycles and the independence number in graphs. Chapter 2 contains problems on independent sets of cycles. It is known that it is hard to compute the maximum cardinality of edge-disjoint and vertex-disjoint cycle packings, even if restricted to subcubic graphs. Therefore, the first section discusses the complexity of a simpler problem: packing cycles of fixed length l in graphs of maximum degree Delta. The results of Caprara and Rizzi, who have solved this problem for l=3 are generalised to arbitrary lengths. The second section describes the structure of graphs for which the edge-disjoint resp. vertex-disjoint cycle packing number differs from the cyclomatic number by a constant. The corresponding classes of 2-connected graphs can be obtained by a simple extension rules applied to a finite set of graphs. This result implies a fixed-parameter-tractability result for the edge-disjoint and vertex-disjoint cycle packing numbers. Chapter 3 contains an approximation of the minimum number of cycle lengths in a Hamiltonian graph with q chords. A family of examples shows that no more than the square root of q+1 can be guaranteed. The main result is that the square root of 4/7*q cycle lengths can be guaranteed. The proof relies on a lemma by Faudree et al., which states that the graph that contains a path with endvertices x and y and q chords of equal length contains paths between x and y of at least q/3 different lengths. The first section corrects the originally faulty proof and derives additional bounds. The second section uses these bounds to derive the lower bound on the size of the cycle spectrum. Chapter 4 focuses on lower bounds on the independence ratio, i.e. the quotient of independence number and order of a graph, for graphs of given density. In the introduction, best-possible bounds both for arbitrary graphs and large connected graphs are derived from known results. Therefore, the rest of this chapter considers classes of graphs defined by forbidding small odd cycles as subgraphs. The main result of the first section is a generalisation of a result of Heckman and Thomas that determines the best possible lower bound for connected triangle-free graphs with average degree at most 10/3 and characterises the extremal graphs. The rest of the chapter is devoted to conjectures with similar statements on connected triangle-free graphs of average degree in [10/3, 54/13] and on connected graphs of odd girth 7 with average degree up to 14/5, similar conjectures for the bipartite ratio, possible classes of extremal graphs for these conjectures, and observations in support of the conjectures