Related papers: A graph theoretic encoding of Lucas sequences
We introduce a new class of auto-encoders for directed graphs, motivated by a direct extension of the Weisfeiler-Leman algorithm to pairs of node labels. The proposed model learns pairs of interpretable latent representations for the nodes…
Finding the maximum number of induced cycles of length $k$ in a graph on $n$ vertices has been one of the most intriguing open problems of Extremal Graph Theory. Recently Balogh, Hu, Lidick\'{y} and Pfender answered the question in the case…
A $3$-uniform loose cycle, denoted by $C_t$, is a $3$-graph on $t$ vertices whose vertices can be arranged cyclically so that each hyperedge consists of three consecutive vertices, and any two consecutive hyperedges share exactly one…
A graph that is completely determined by its degree sequence is called a unigraph. In 2000, Regina Tyshkevich published one of the most important papers on unigraphs. There are two parts to the paper: a decomposition theorem that describes…
We study combinatorial indicators related to the characteristic phase transitions associated with coloring a graph optimally and finding a maximum independent set. In particular, we investigate the role of the acyclic orientations of the…
In earlier work we introduced the graph bracket polynomial of graphs with marked vertices, motivated by the fact that the Kauffman bracket of a link diagram D is determined by a looped, marked version of the interlacement graph associated…
The vertices of a $k$-token graph of a graph $G$ correspond to $k$ indistinguishable tokens placed on $k$ different vertices of $G$. Changing some conditions on both the nature of the tokens and the number of tokens allowed in each vertex…
In a digraph $D=(V,A)$, an oriented path is a sequence $P=x_1x_2\dots x_p$ of distinct vertices such that either $x_ix_{i+1}\in A$ or $x_{i+1}x_{i}\in A$ or both for every $i\in [p-1]$. If $x_ix_{i+1}\in A$ in $P$, then $x_ix_{i+1}$ is a…
Supersingular elliptic curve isogeny graphs underlie isogeny-based cryptography. For isogenies of a single prime degree $\ell$, their structure has been investigated graph-theoretically. We generalise the notion of $\ell$-isogeny graphs to…
An edge of a graph of order $n$ is pancyclic if it lies in a cycle of every length $3,\ldots,n$. A graph of order $n$ is vertex-pancyclic if every vertex lies in a cycle of every length $3,\ldots,n$. Recently, Li and Zhan proved that every…
In any network, the interconnection of nodes by means of geodesics and the number of geodesics existing between nodes are important. There exists a class of centrality measures based on the number of geodesics passing through a vertex.…
We are concerned with split graphs and pseudo-split graphs whose complements are isomorphic to themselves. These special subclasses of self-complementary graphs are actually the core of self-complementary graphs. Indeed, we show that all…
A theorem of Ding, Oporowski, Oxley, and Vertigan implies that any sufficiently large twin-free graph contains a large matching, a co-matching, or a half-graph as a semi-induced subgraph. The sizes of these unavoidable patterns are measured…
A hole in a graph is a chordless cycle of length at least 4. A theta is a graph formed by three internally vertex-disjoint paths of length at least 2 between the same pair of distinct vertices. A wheel is a graph formed by a hole and a node…
Motivated to find the answers to some of the questions that have occurred in recent papers dealing with Hamiltonian cycles (abbreviated HCs) in some special classes of grid graphs we started the investigation of spanning unions of cycles,…
We provide four equivalent combinatorial conditions for a simple assembly graph (rigid vertex graph where all vertices are of degree 1 or 4) to have the largest number of Hamiltonian sets of polygonal paths relative its size. These…
Computing maximum independent sets in graphs is an important problem in computer science. In this paper, we develop an evolutionary algorithm to tackle the problem. The core innovations of the algorithm are very natural combine operations…
We study $t$-designs of parameters $(n,k,\lambda)$ over finite fields as group divisible designs and set systems admitting a transitive action of a linear group encoded in an hypergraph $G$ whose vertex set of size $n$ is partitioned into…
The vertices of a graph are classified into seven types by J.T. Hedetniemi, S.M. Hedetniemi, S.T. Hedetniemi and T.M. Lewis and they ask the following questions: 1) What is the smallest order $n$ of a graph having $n-2$ very typical…
It is well known that the set of possible degree sequences for a graph on $n$ vertices is the intersection of a lattice and a convex polytope. We show that the set of possible degree sequences for a $k$-uniform hypergraph on $n$ vertices is…