Related papers: On the maximum degree of path-pairable planar grap…

It is proved that every series-parallel digraph whose maximum vertex-degree is $\Delta$ admits an upward planar drawing with at most one bend per edge such that each edge segment has one of $\Delta$ distinct slopes. This is shown to be…

A graph is $1$-planar if it has a drawing in the plane such that each edge is crossed at most once by another edge. Moreover, if this drawing has the additional property that for each crossing of two edges the end vertices of these edges…

Let $n, s$ be positive integers such that $n$ is sufficiently large and $s\le n/3$. Suppose $H$ is a 3-uniform hypergraph of order $n$. If $H$ contains no isolated vertex and $deg(u)+ deg(v) > 2(s-1)(n-1)$ for any two vertices $u$ and $v$…

A strong edge-colouring of a graph is a proper edge-colouring where each colour class induces a matching. It is known that every planar graph with maximum degree $\Delta$ has a strong edge-colouring with at most $4\Delta+4$ colours. We show…

A path cover is a decomposition of the edges of a graph into edge-disjoint simple paths. Gallai conjectured that every connected $n$-vertex graph has a path cover with at most $\lceil n/2 \rceil$ paths. We prove Gallai's conjecture for…

The planar Tur\'{a}n number of a given graph $H$, denoted by $ex_{\mathcal{P}}(n,H)$, is the maximum number of edges over all planar graphs on $n$ vertices that do not contain a copy of $H$ as a subgraph. Let $H_k$ be a friendship graph,…

A graph $G$ is $k$-path-coverable if its vertex set $V(G)$ can be covered by $k$ or fewer vertex disjoint paths. In this paper, using the $Q$-index of a connected graph $G$, we present a tight sufficient condition for $G$ with fixed minimum…

In the branch of mathematics known as graph theory, graphs are considered as a set of points, called vertices, with connections between these points, called edges. The purpose of this paper is to study mappings between two graphs that have…

A graph is $1$-$planar$ if it can be drawn in the plane so that each edge is crossed by at most one other edge. Moreover, a 1-planar graph $G$ is $optimal$ if it satisfies $|E(G)|=4|V(G)|-8$. J. Fujisawa et al. [16] first considered…

We give sufficient conditions under which a random graph with a specified degree sequence is symmetric or asymmetric. In the case of bounded degree sequences, our characterisation captures the phase transition of the symmetry of the random…

For large $n$ we determine the maximum number of induced 6-cycles which can be contained in a planar graph on $n$ vertices, and we classify the graphs which achieve this maximum. In particular we show that the maximum is achieved by the…

A perfect matching in a hypergraph is a set of edges that partition the set of vertices. We study the complexity of deciding the existence of a perfect matching in orderable and separable hypergraphs. We show that the class of orderable…

Consider the family of all finite graphs with maximum degree $\Delta(G)<d$ and matching number $\nu(G)<m$. In this paper we give a new proof to obtain the exact upper bound for the number of edges in such graphs and also characterize all…

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…

We show that for every l, there exists d_l such that every 3-edge-connected graph with minimum degree d_l can be edge-partitioned into paths of length l (provided that its number of edges is divisible by l). This improves a result asserting…

Let f(k) denote the maximum such that every simple undirected graph containing two vertices s,t and k edge-disjoint s-t paths, also contains two vertices u,v and f(k) independent u-v paths. Here, a set of paths is independent if none of…

A vertex colouring of a graph is \emph{nonrepetitive on paths} if there is no path $v_1,v_2,...,v_{2t}$ such that v_i and v_{t+i} receive the same colour for all i=1,2,...,t. We determine the maximum density of a graph that admits a…

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…

A graph is called 1-planar if there exists its drawing in the plane such that each edge is crossed at most once. In this paper, we study 1-planar graph joins. We prove that the join $G+H$ is 1-planar if and only if the pair $[G,H]$ is…

The twisted graph $T_{n}$ is a drawing of the complete graph with $n$ vertices $v_{1},v_{2},\ldots ,v_{n}$ in which two edges $v_{i}v_{j}$ ($i<j$) and $v_{s}v_{t}$ ($s<t$) cross if and only if $i<s<t<j$ or $s<i<j<t$. We show that for any…