Related papers: Counting degree-constrained subgraphs and orientat…
The large $N$ reductions in gauge theories are identified with dimensional reductions with homogeneous distribution of the eigenvalues of the gauge field, and it is used to identify the corresponding closed string descriptions in the…
Let $G=(V,E)$ be a finite, simple, connected, combinatorial graph on $n$ vertices and let $D \in \mathbb{R}^{n \times n}$ be its graph distance matrix $D_{ij} = d(v_i, v_j)$. Steinerberger (J. Graph Theory, 2023) empirically observed that…
Let $G$ be a finite, simple, and undirected graph of order $n$ and average degree $d$. Up to terms of smaller order, we characterize the minimal intervals $I$ containing $d$ that are guaranteed to contain some vertex degree. In particular,…
A graph $G$ is called collapsible if for every even subset $R\subseteq V(G)$, there is a spanning connected subgraph $H$ of $G$ such that $R$ is the set of vertices of odd degree in $H$. A graph is the reduction of $G$ if it is obtained…
Graphlets are subgraphs rooted at a fixed vertex. The number of occurrences of graphlets aligned to a particular vertex, called graphlet degree sequence (gds), gives a topological description of the surrounding of the analyzed vertex.…
Haj\'os' conjecture states that an Eulerian graph of order n can be decomposed into at most (n-1)/2 edge-disjoint cycles. We describe preprocessing steps, heuristics and integer programming techniques that enable us to verify Haj\'os'…
The distinguishing number $\operatorname D(G)$ of a graph $G$ is the least cardinal $d$ such that $G$ has a labeling with $d$ labels which is only preserved by the trivial automorphism. We show that the distinguishing number of infinite,…
We study how eigenvectors of random regular graphs behave when projected onto fixed directions. For a random $d$-regular graph with $N$ vertices, where the degree $d$ grows slowly with $N$, we prove that these projections follow…
Given a graphical degree sequence ${\bf d}=(d_1,\ldots, d_n)$, let $G(n, {\bf d})$ denote a uniformly random graph on vertex set $[n]$ where vertex $ i$ has degree $d_i$ for every $1\le i\le n$. We give upper and lower bounds on the joint…
Given two $k$-dicolourings of a digraph $D$, we prove that it is PSPACE-complete to decide whether we can transform one into the other by recolouring one vertex at each step while maintaining a dicolouring at any step even for $k=2$ and for…
Given a graph $G$, a decomposition of $G$ is a partition of its edges. A graph is $(d, h)$-decomposable if its edge set can be partitioned into a $d$-degenerate graph and a graph with maximum degree at most $h$. For $d \le 4$, we are…
In this paper, we first generalize a theorem for counting the number of faces of an oriented embedding of a graph that passing through a given cut-edge set [S. Stahl, Trans. Amer. Math. Soc. 259 (1980), 129--145] to all surfaces. Then we…
In this paper, we present enumeration algorithms to list all preferred extensions of an argumentation framework. This task is equivalent to enumerating all maximal semikernels of a directed graph. For directed graphs on $n$ vertices, all…
The Erd\H{o}s, Gr\"unwald, and Weiszfeld theorem is a characterization of those infinite graphs which are Eulerian. That is, infinite graphs that admit infinite Eulerian paths. In this article we prove an effective version of the Erd\H{o}s,…
The well-known 1-2-3 Conjecture asserts that the edges of every graph without an isolated edge can be weighted with $1$, $2$ and $3$ so that adjacent vertices receive distinct weighted degrees. This is open in general. We prove that every…
The twin-width of a graph $G$ is the minimum integer $d$ such that $G$ has a $d$-contraction sequence, that is, a sequence of $|V(G)|-1$ iterated vertex identifications for which the overall maximum number of red edges incident to a single…
A well known Euler's formula consequence's corollary in graph theory states that: For a connected simple planar graph with $n$ vertices and $m$ edges, and girth $g$, we have $m \leq \frac{g}{g-2}(n-2)$. We show that a connected simple plane…
Modular graph forms are a class of modular covariant functions which appear in the genus-one contribution to the low-energy expansion of closed string scattering amplitudes. Modular graph forms with holomorphic subgraphs enjoy the…
In 1985, Mader conjectured the existence of a function $f$ such that every digraph with minimum out-degree at least $f(k)$ contains a subdivision of the transitive tournament of order $k$. This conjecture is still completely open, as the…
We prove that for each $d\geq 3$ and $k\geq 2$, the set of limit points of the first $k$ eigenvalues of sequences of $d$-regular graphs is \[ \{(\mu_1,\dots,\mu_k): d=\mu_1\geq \dots\geq \mu_{k}\geq2\sqrt{d-1}\}. \] The result for $k=2$ was…