Related papers: A counterexample regarding labelled well-quasi-ord…
Robertson and Seymour proved that the relation of graph immersion is well-quasi-ordered for finite graphs. Their proof uses the results of graph minors theory. Surprisingly, there is a very short proof of the corresponding rough structure…
A normal Helly circular-arc graph is the intersection graph of arcs on a circle of which no three or less arcs cover the whole circle. Lin, Soulignac, and Szwarcfiter [Discrete Appl. Math. 2013] characterized circular-arc graphs that are…
A graph is well-covered if all its maximal independent sets are of the same cardinality (Plummer, 1970). If G is a well-covered graph, has at least two vertices, and G-v is well-covered for every vertex v, then G is a 1-well-covered graph…
Chordal graphs are the graphs in which every cycle of length at least four has a chord. A set $S$ is a vertex separator for vertices $a$ and $b$ if the removal of $S$ of the graph separates $a$ and $b$ into distinct connected components. A…
Let $G$ be a simple graph with maximum degree $\Delta(G)$. A subgraph $H$ of $G$ is overfull if $|E(H)|>\Delta(G)\lfloor \frac{1}{2}|V(H)| \rfloor$. Chetwynd and Hilton in 1986 conjectured that a graph $G$ with $\Delta(G)>\frac{1}{3}|V(G)|$…
A graph is almost bipartite if it contains exactly one odd cycle, and it is Konig-Egervary if the sum of the independence number and the matching number equals the order of the graph. We introduce the class of Bipartite-Almost Bipartite…
Chordal graphs are important in algorithmic graph theory. Chordal digraphs are a digraph analogue of chordal graphs and have been a subject of active studies recently. Unlike chordal graphs, chordal digraphs lack many structural properties…
Mock threshold graphs are a simple generalization of threshold graphs that, like threshold graphs, are perfect graphs. Our main theorem is a characterization of mock threshold graphs by forbidden induced subgraphs. Other theorems…
We extend the notion of quasi-transitive orientations of graphs to 2-edge-coloured graphs. By relating quasi-transitive $2$-edge-colourings to an equivalence relation on the edge set of a graph, we classify those graphs that admit a…
A set S of vertices in a graph G is a dominating set of G if every vertex not in S is adjacent to a vertex in S . The domination number of G, denoted by $\gamma$(G), is the minimum cardinality of a dominating set in G. In a breakthrough…
Huang proved that every set of more than half the vertices of the $d$-dimensional hypercube $Q_d$ induces a subgraph of maximum degree at least $\sqrt{d}$, which is tight by a result of Chung, F\"uredi, Graham, and Seymour. Huang asked…
The aim of this paper is to apply the framework, which was developed by Sam and Snowden, to study structural properties of graph homologies, in the spirit of Ramos, Miyata and Proudfoot. Our main results concern the magnitude homology of…
We prove a new generalisation of Ramsey's theorem by showing that every $2$-edge-coloured graph with sufficiently large minimum degree contains a monochromatic induced subgraph whose minimum degree remains large. From this, we also derive…
A graph $G$ is said to be $p$-locally dense if every induced subgraph of $G$ with linearly many vertices has edge density at least $p$. A famous conjecture of Kohayakawa, Nagle, R\"odl, and Schacht predicts that locally dense graphs have,…
We generalize the notion of quasirandom which concerns a class of equivalent properties that random graphs satisfy. We show that the convergence of a graph sequence under the spectral distance is equivalent to the convergence using the…
Sidorenko's Conjecture asserts that every bipartite graph H has the Sidorenko property, i.e., a quasirandom graph minimizes the density of H among all graphs with the same edge density. We study a stronger property, which requires that a…
In this article, we introduce a relation including ideals of an evolution algebra and hereditary subsets of vertices of its associated graph and establish some properties among them. This relation allows us to determine maximal ideals and…
A coprime labeling of a graph of order $n$ is an assignment of distinct positive integer labels in which adjacent vertices have relatively prime labels. Restricting labels to only the set $1$ to $n$ results in a prime labeling. In this…
A simple undirected graph is said to be {\em semisymmetric} if it is regular and edge-transitive but not vertex-transitive. Every semisymmetric graph is a bipartite graph with two parts of equal size. It was proved in [{\em J. Combin.…
In this manuscript we develop a version of Szemer\'edi's regularity lemma that is suitable for analyzing multicolorings of complete graphs and directed graphs. In this, we follow the proof of Alon, Fischer, Krivelevich and M. Szegedy…