Related papers: Maximum 4-degenerate subgraph of a planar graph

In this paper we raise a variant of a classic problem in extremal graph theory, which is motivated by a design of fractional repetition codes, a model in distributed storage systems. For any feasible positive integers $d\geq 3$, $n \geq 3$,…

The Alon-Tarsi number of a graph $ G $ is the smallest $ k $ such that there exists an orientation $ D $ of $ G $ with maximum outdegree $ k - 1 $ satisfying that the number of even Eulerian subgraphs is different from the number of odd…

The 'separation dimension' of a graph $G$ is the smallest natural number $k$ for which the vertices of $G$ can be embedded in $\mathbb{R}^k$ such that any pair of disjoint edges in $G$ can be separated by a hyperplane normal to one of the…

A vertex whose removal in a graph $G$ increases the number of components of $G$ is called a cut vertex. For all $n,c$, we determine the maximum number of connected induced subgraphs in a connected graph with order $n$ and $c$ cut vertices,…

The reconfiguration graph $R_k(G)$ for the $k$-colorings of a graph $G$ has as vertex set the set of all possible proper $k$-colorings of $G$ and two colorings are adjacent if they differ in the color of exactly one vertex. A result of…

A graph of order $n$ is said to be $k$-\emph{factor-critical} $(0\le k<n)$ if the removal of any $k$ vertices results in a graph with a perfect matching. A $k$-factor-critical graph $G$ is \emph{minimal} if $G-e$ is not $k$-factor-critical…

A connected graph $G$ is said to be $k$-connected if it has more than $k$ vertices and remains connected whenever fewer than $k$ vertices are deleted. In this paper, for a connected graph $G$ with sufficiently large order, we present a…

A graph $G$ of order $n$ is said to be $k$-factor-critical for integers $1\leq k < n$, if the removal of any $k$ vertices results in a graph with a perfect matching. $1$- and $2$-factor-critical graphs are the well-known factor-critical and…

We prove that, for every positive integer k, there is an integer N such that every 4-connected non-planar graph with at least N vertices has a minor isomorphic to K_{4,k}, the graph obtained from a cycle of length 2k+1 by adding an edge…

A graph on $n \ge 3$ vertices drawn in the plane such that each edge is crossed at most four times has at most $6(n-2)$ edges -- this result proven by Ackerman is outstanding in the literature of beyond-planar graphs with regard to its…

A graph is $k$-planar if it can be drawn in the plane such that no edge is crossed more than $k$ times. While for $k=1$, optimal $1$-planar graphs, i.e., those with $n$ vertices and exactly $4n-8$ edges, have been completely characterized,…

A digraph $D$ is an oriented graph if $D$ does not have a pair of opposite arcs. The degree of a vertex $v$ of $D$ is the sum of the in-degree and out-degree of $v.$ Let $fvs(D)$ be the minimum number of vertices whose deletion from $D$…

Let $G$ be a graph and $v$ any vertex of $G$. We define the degenerate degree of $v$, denoted by $\zeta(v)$ as $\zeta(v)={\max}_{H: v\in H}~\delta(H)$, where the maximum is taken over all subgraphs of $G$ containing the vertex $v$. We show…

We prove several results on coloring squares of planar graphs without 4-cycles. First, we show that if $G$ is such a graph, then $G^2$ is $(\Delta(G)+72)$-degenerate. This implies an upper bound of $\Delta(G)+73$ on the chromatic number of…

A strong edge-coloring of a graph $G$ is an edge-coloring in which every color class is an induced matching, and the strong chromatic index $\chi_s'(G)$ is the minimum number of colors needed in strong edge-colorings of $G$. A graph is…

An odd graph is a finite graph all of whose vertices have odd degrees. Given graph $G$ is decomposable into $k$ odd subgraphs if its edge set can be partitioned into $k$ subsets each of which induces an odd subgraph of $G$. The minimum…

We call a finite undirected graph minimally k-matchable if it has at least k distinct perfect matchings but deleting any edge results in a graph which has not. An odd subdivision of some graph G is any graph obtained by replacing every edge…

We prove that one can perfectly pack degenerate graphs into complete or dense $n$-vertex quasirandom graphs, provided that all the degenerate graphs have maximum degree $o(\frac{n}{\log n})$, and in addition $\Omega(n)$ of them have at most…

The kth power of a simple graph G, denoted G^k, is the graph with vertex set V(G) where two vertices are adjacent if they are within distance k in G. We are interested in finding lower bounds on the average degree of G^k. Here we prove that…

We prove that every 3-connected planar graph on $n$ vertices contains an induced path on $\Omega(\log n)$ vertices, which is best possible and improves the best known lower bound by a multiplicative factor of $\log \log n$. We deduce that…