Related papers: The maximum number of maximum dissociation sets in…

An identifying code $C$ of a graph $G$ is a dominating set of $G$ such that any two distinct vertices of $G$ have distinct closed neighbourhoods within $C$. These codes have been widely studied for over two decades. We give an improvement…

The pseudoforest version of the Strong Nine Dragon Tree Conjecture states that if a graph $G$ has maximum average degree $\text{mad}(G) = 2 \max_{H \subseteq G} \frac{e(G)}{v(G)}$ at most $2(k + \frac{d}{k+d+1})$, then it has a…

Let $\binom{[n]}{k}$ denote the collection of all $k$-subsets of the standard $n$-set $[n]=\{1,2,\ldots,n\}$. Let $n>2k$ and let $\mathcal{F}\subset \binom{[n]}{k}$ be an {\it intersecting} $k$-graph, i.e., $F\cap F'\neq \emptyset$ for all…

The Graceful Tree Conjecture of Rosa from 1967 asserts that the vertices of each tree T of order n can be injectively labelled by using the numbers {1,2,...,n} in such a way that the absolute differences induced on the edges are pairwise…

An $\textit{identifying code}$ of a closed-twin-free graph $G$ is a set $S$ of vertices of $G$ such that any two vertices in $G$ have a distinct intersection between their closed neighborhood and $S$. It was conjectured that there exists a…

The sigma-irregularity index $\sigma(G) = \sum_{uv \in E(G)} (d_G(u) - d_G(v))^2$ measures the total degree imbalance along the edges of a graph. We study extremal problems for $\sigma(T)$ within the class of trees of fixed order $n$ and…

A graph $G$ is $D$-distinguishable if there is a labeling of its vertices with $D$ labels such that the only automorphism of $G$ which preserves the labeling is the identity. The distinguishing number of $G$ is the minimum value $D$ for…

In this paper, we show that every planar graph without $4$-cycles and $6$-cycles has a partition of its vertex set into two sets, where one set induces a forest, and the other induces a forest with maximum degree at most $2$ (equivalently,…

We explore the concept of separating systems of vertex sets of graphs. A separating system of a set $X$ is a collection of subsets of $X$ such that for any pair of distinct elements in $X$, there exists a set in the separating system that…

We show that the maximum number of minimum dominating sets of a forest with domination number $\gamma$ is at most $\sqrt{5}^{\gamma}$ and construct for each $\gamma$ a tree with domination number $\gamma$ that has more than…

Given a positive integer $n$, an $n$-ladder is a lower finite lattice whose elements have at most $n$ lower covers. In 1984, Ditor proved that every $n$-ladder has cardinality at most $\aleph_{n-1}$ and asked whether this bound is sharp,…

We study the maximal Steklov eigenvalues of trees with given number of boundary vertices and total number of vertices. Trees can be regarded as discrete analogue of Hadamard manifolds, namely simply-connected Riemannian manifolds of…

We give deterministic distributed $(1+\epsilon)$-approximation algorithms for Minimum Vertex Coloring and Maximum Independent Set on chordal graphs in the LOCAL model. Our coloring algorithm runs in $O(\frac{1}{\epsilon} \log n)$ rounds,…

Let $\mathcal{O}_n$ be the set of ordered labeled trees on ${0,...,n}$. A maximal decreasing subtree of an ordered labeled tree is defined by the maximal ordered subtree from the root with all edges being decreasing. In this paper, we study…

Tree-decompositions of graphs are of fundamental importance in structural and algorithmic graph theory. The main property of tree-decompositions is the width (the maximum size of a bag minus 1). We show that every graph has a…

We continue the study of $(\mathrm{tw},\omega)$-bounded graph classes, that is, hereditary graph classes in which the treewidth can only be large due to the presence of a large clique, with the goal of understanding the extent to which this…

Many interesting problems are obtained by attempting to generalize classical results on convexity in Euclidean spaces to other convexity spaces, in particular to convexity spaces on graphs. In this paper we consider $P_3$-convexity on…

Let $G(V,E)$ be a finite, simple, isolate-free graph. Two disjoint sets $A,B\subset V$ form a total coalition in $G$, if none of them is a total dominating set, but their union $A\cup B$ is a total dominating set. A vertex partition…

A simple graph $G$ is an {\it 2-tree} if $G=K_3$, or $G$ has a vertex $v$ of degree 2, whose neighbors are adjacent, and $G-v$ is an 2-tree. Clearly, if $G$ is an 2-tree on $n$ vertices, then $|E(G)|=2n-3$. A non-increasing sequence…

A $k$-block in a graph $G$ is a maximal set of at least $k$ vertices no two of which can be separated in $G$ by removing less than $k$ vertices. It is separable if there exists a tree-decomposition of adhesion less than $k$ of $G$ in which…