Related papers: Minimum degree, independence number and pseudo [2,…

The $k$-core of a graph is the largest subgraph of minimum degree at least $k$. We show that for $k$ sufficiently large, the $(k + 2)$-core of a random graph $\G(n,p)$ asymptotically almost surely has a spanning $k$-regular subgraph. Thus…

A graph $G$ is a fractional $(a,b,k)$-critical covered graph if $G-U$ is a fractional $[a,b]$-covered graph for every $U\subseteq V(G)$ with $|U|=k$, which is first defined by Zhou, Xu and Sun (S. Zhou, Y. Xu, Z. Sun, Degree conditions for…

A graph is called an $(r,k)$-graph if its vertex set can be partitioned into $r$ parts, each having at most $k$ vertices and there is at least one edge between any two parts. Let $f(r,H)$ be the minimum $k$ for which there exists an…

An identifying code is a subset of vertices of a graph such that each vertex is uniquely determined by its neighbourhood within the identifying code. If $\M(G)$ denotes the minimum size of an identifying code of a graph $G$, it was…

A 2-hued coloring of a graph $G$ (also known as conditional $(k, 2)$-coloring and dynamic coloring) is a coloring such that for every vertex $v\in V(G)$ of degree at least $2$, the neighbors of $v$ receive at least $2$ colors. The smallest…

Given a proper edge coloring $\varphi$ of a graph $G$, we define the palette $S_{G}(v,\varphi)$ of a vertex $v \in V(G)$ as the set of all colors appearing on edges incident with $v$. The palette index $\check s(G)$ of $G$ is the minimum…

In this paper we introduce and study a new graph invariant derived from the degree sequence of a graph $G$, called the sub-$k$-domination number and denoted $sub_k(G)$. We show that $sub_k(G)$ is a computationally efficient sharp lower…

For a graph $G$, the spectral radius of $G$ is the largest eigenvalue of its adjacency matrix. A connected factor of $G$ is a connected spanning subgraph of $G$. For example, a spanning tree of $G$ is a 1-connected factor of $G$. Let $G$ be…

We say that $G$ is a $(3, 3)$-Ramsey graph if every $2$-coloring of the edges of $G$ forces a monochromatic triangle. The $(3, 3)$-Ramsey graph $G$ is minimal if $G$ does not contain a proper $(3, 3)$-Ramsey subgraph. In this work we find…

The distinguishing index of a simple graph $G$, denoted by $D'(G)$, is the least number of labels in an edge labeling of $G$ not preserved by any non-trivial automorphism. It was conjectured by Pil\'sniak (2015) that for any 2-connected…

Extremal problems involving the enumeration of graph substructures have a long history in graph theory. For example, the number of independent sets in a $d$-regular graph on $n$ vertices is at most $(2^{d+1}-1)^{n/2d}$ by the Kahn-Zhao…

A cograph is a simple graph which contains no path on 4 vertices as an induced subgraph. We consider the eigenvalues of adjacency matrices of cographs and prove that a graph $G$ is a cograph if and only if no induced subgraph of $G$ has an…

A subset $S\subseteq V$ in a graph $G=(V,E)$ is a total $[1,2]$-set if, for every vertex $v\in V$, $1\leq |N(v)\cap S|\leq 2$. The minimum cardinality of a total $[1,2]$-set of $G$ is called the total $[1,2]$-domination number, denoted by…

A {\it 2-rainbow domination function} of a graph $G$ is a function $f$ that assigns to each vertex a set of colors chosen from the set $\{1,2\}$, such that for any $v\in V(G)$, $f(v)=\emptyset$ implies $\bigcup_{u\in N(v)}f(u)=\{1,2\}$. The…

A subgraph $H$ of a multigraph $G$ is called strongly spanning, if any vertex of $G$ is not isolated in $H$, while it is called maximum $k$-edge-colorable, if $H$ is proper $k$-edge-colorable and has the largest size. We introduce a…

Let $G$ be a finite group. Denoting by ${\rm{cd}}(G)$ the set of the degrees of the irreducible complex characters of $G$, we consider the {\it character degree graph} of $G$: this is the (simple, undirected) graph whose vertices are the…

For a graph $G$, its $k$-th power $G^k$ is constructed by placing an edge between two vertices if they are within distance $k$ of each other. The $k$-independence number $\alpha_k(G)$ is defined as the independence number of $G^k$. By using…

For an edge-colored graph $G$, the minimum color degree of $G$ means the minimum number of colors on edges which are adjacent to each vertex of $G$. We prove that if $G$ is an edge-colored graph with minimum color degree at least $5$ then…

A bisection of a graph is a bipartition of its vertex set in which the number of vertices in the two parts differ by at most 1, and its size is the number of edges which go across the two parts. In this paper, motivated by several questions…

A graph G is a multi-interval PCG if there exist an edge weighted tree T with non-negative real values and disjoint intervals of the non-negative real half-line such that each node of G is uniquely associated to a leaf of T and there is an…