Related papers: The Burning Number Conjecture Holds Asymptotically

The famous List Colouring Conjecture from the 1970s states that for every graph $G$ the chromatic index of $G$ is equal to its list chromatic index. In 1996 in a seminal paper, Kahn proved that the List Colouring Conjecture holds…

Let $G$ be a connected graph of order $n$ with vertex set $V(G)$. A subset $S\subseteq V(G)$ is an $(a,b)$-dominating set if every vertex $v\in S$ is adjacent to at least $a$ vertices in $S$ and every $v\in V\setminus S$ is adjacent to at…

The bandwidth theorem [Mathematische Annalen, 343(1):175--205, 2009] states that any $n$-vertex graph $G$ with minimum degree $\big(\tfrac{k-1}{k}+o(1)\big)n$ contains all $n$-vertex $k$-colourable graphs $H$ with bounded maximum degree and…

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 Brouwer conjecture (BC) in spectral graph theory claims that the sum of the largest k Kirchhoff eigenvalues of a graph are bounded above by the number m of edges plus k(k+1)/2. We show that (BC) holds for all graphs with n vertices if n…

A path in an edge-colored graph is called proper if no two consecutive edges of the path receive the same color. For a connected graph $G$, the proper connection number $pc(G)$ of $G$ is defined as the minimum number of colors needed to…

The Tur\'{a}n number $T(n,\alpha+1,r)$ is the minimum number of edges in an $n$-vertex $r$-graph whose independence number does not exceed $\alpha$. For each $r\geq 2$, there exists $t_*(r)$ such that $T(n,\alpha+1,r) = t_*(r) \: n^r \:…

Contraction of triangles is a standard operation in the study of cubic graphs, as it reduces the order of the graph while typically preserving many of its properties. In this paper, we investigate the converse problem, wherein certain…

We consider the number of vertices that must be removed from a graph G in order that the remaining subgraph has no component with more than k vertices. Our principal observation is that, if G is a sparse random graph or a random regular…

Suppose in a graph $G$ vertices can be either red or blue. Let $k$ be odd. At each time step, each vertex $v$ in $G$ polls $k$ random neighbours and takes the majority colour. If it doesn't have $k$ neighbours, it simply polls all of them,…

A well-studied concept is that of the total chromatic number. A proper total colouring of a graph is a colouring of both vertices and edges so that every pair of adjacent vertices receive different colours, every pair of adjacent edges…

We show that every graph $G$ on $n$ vertices with $\delta(G) \geq (1/2+\varepsilon)n$ is spanned by a complete blow-up of a cycle with clusters of nearly uniform size $\Omega(\log n)$. The proof is based on a recently introduced approach…

Let $G$ be a graph of order $n$ with adjacency matrix $A(G)$. The \textit{energy} of graph $G$, denoted by $\mathcal{E}(G)$, is defined as the sum of absolute value of eigenvalues of $A(G)$. It was conjectured that if $A(G)$ is…

We prove that for every graph $G$ on $n$ vertices and with minimum degree five, the domination number $\gamma(G)$ cannot exceed $n/3$. The proof combines an algorithmic approach and the discharging method. Using the same technique, we…

We conjecture that every graph of minimum degree five with no separating triangles and drawn in the plane with one crossing is 4-colorable. In this paper, we use computer enumeration to show that this conjecture holds for all graphs with at…

An adjacent vertex distinguishing coloring of a graph G is a proper edge coloring of G such that any pair of adjacent vertices are incident with distinct sets of colors. The minimum number of colors needed for an adjacent vertex…

Ohba has conjectured \cite{ohb} that if the graph $G$ has $2\chi(G)+1$ or fewer vertices then the list chromatic number and chromatic number of $G$ are equal. In this paper we prove that this conjecture is asymptotically correct. More…

For a graph $G$ with vertex set $V$, let N($G$) denote the number of nonempty subsets of $V$ that induce a connected graph in $G$. In this paper, we focus on determining N($G$) for $G$ in the family $\mathbb{B}_n$ of $n$-vertex bicyclic…

A dominating set of a graph $G$ is a set $D\subseteq V_G$ such that every vertex in $V_G-D$ is adjacent to at least one vertex in $D$, and the domination number $\gamma(G)$ of $G$ is the minimum cardinality of a dominating set of $G$. A set…

Graph pebbling is a network optimization model for satisfying vertex demands with vertex supplies (called pebbles), with partial loss of pebbles in transit. The pebbling number of a demand in a graph is the smallest number for which every…