Related papers: Gallai's path decomposition conjecture for graphs …

An edge coloring of the $n$-vertex complete graph, $K_n$, is a Gallai coloring if it does not contain any rainbow triangle, that is, a triangle whose edges are colored with three distinct colors. We prove that for $n$ large and every $k$…

We prove that the maximal number of directed edges in a vertex-critical strongly connected simple digraph on n vertices is n(n-1)/2 - n +4.

The classic theorem of Gallai and Milgram (1960) generalizes several fundamental results in Graph Theory, such as Dilworth's theorem on posets and K\H{o}nig's theorem on matchings in bipartite graphs. The theorem asserts that for every…

Let $i_t(G)$ be the number of independent sets of size $t$ in a graph $G$. Engbers and Galvin asked how large $i_t(G)$ could be in graphs with minimum degree at least $\delta$. They further conjectured that when $n\geq 2\delta$ and $t\geq…

We show that the edges of any planar graph of maximum degree at most $9$ can be partitioned into $4$ linear forests and a matching. Combined with known results, this implies that the edges of any planar graph $G$ of odd maximum degree…

We give bounds on the L(2,1)-labeling number of a simple graph in terms of its order and its maximum degree. We also describe an infinite class of graphs of which the elements have the highest L(2,1)-labeling numbers in terms of their…

A graph is \emph{hamiltonian-connected} if every pair of vertices can be connected by a hamiltonian path, and it is \emph{hamiltonian} if it contains a hamiltonian cycle. We construct families of non-hamiltonian graphs for which the ratio…

It was independently conjectured by H\"aggkvist in 1989 and Kriesell in 2011 that given a positive integer $\ell$, every simple eulerian graph with high minimum degree (depending on $\ell$) admits an eulerian tour such that every segment of…

In this paper we investigate results of the form "every graph $G$ has a cycle $C$ such that the induced subgraph of $G$ on $V(G)\setminus V(C)$ has small maximum degree." Such results haven't been studied before, but are motivated by the…

A tuple (s1,t1,s2,t2) of vertices in a simple undirected graph is 2-linked when there are two vertex-disjoint paths respectively from s1 to t1 and s2 to t2. A graph is 2-linked when all such tuples are 2-linked. We give a new and simple…

For a simple graph $G$, let $n$ and $m$ denote the number of vertices and edges in $G$, respectively. The Erd\H{o}s-Gallai theorem for paths states that in a simple $P_k$-free graph, $m \leq \frac{n(k-1)}{2}$, where $P_k$ denotes a path…

A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this note we give examples of class two 1-planar graphs with maximum degree six or seven.

A vertex coloring of a graph G is called a 2-distance coloring if any two vertices at a distance at most 2 from each other receive different colors. Suppose that G is a planar graph with a maximum degree at most 5. We prove that G admits a…

A {\em $(d,h)$-decomposition} of a graph $G$ is an order pair $(D,H)$ such that $H$ is a subgraph of $G$ where $H$ has the maximum degree at most $h$ and $D$ is an acyclic orientation of $G-E(H)$ of maximum out-degree at most $d$. A graph…

In this paper, we bound the number of edges of a maximal permutation graph with n vertices. We propose a new method to compute the lower bound by splitting the set of labellings of the edges into six parts, considering one separate problem…

We apply the Discharging Method to prove the 1,2,3-Conjecture and the 1,2-Conjecture for graphs with maximum average degree less than 8/3. Stronger results on these conjectures have been proved, but this is the first application of…

We prove that for all graphs with at most $(3.75-o(1))n$ edges there exists a 2-coloring of the edges such that every monochromatic path has order less than $n$. This was previously known to be true for graphs with at most $2.5n-7.5$ edges.…

We prove that a large family of graphs which are decomposable with respect to the modular decomposition can be reconstructed from their collection of vertex-deleted subgraphs.

Recently, variants of many classical extremal theorems have been proved in the random environment. We, complementing existing results, extend the Erd\H{o}s-Gallai Theorem in random graphs. In particular, we determine, up to a constant…

We study how many comparability subgraphs are needed to partition the edge set of a perfect graph. We show that many classes of perfect graphs can be partitioned into (at most) two comparability subgraphs and this holds for almost all…