Related papers: A note on highly connected $K_{2,\ell}$-minor free…

We characterize classes of graphs closed under taking vertex-minors and having no $P_n$ and no disjoint union of $n$ copies of the $1$-subdivision of $K_{1,n}$ for some $n$. Our characterization is described in terms of a tree of radius $2$…

We demonstrate that for every positive integer $\Delta$, every K\_4-minor-free graph with maximum degree $\Delta$ admits an equitable coloring with k colors wherek $\ge$ ($\Delta$+3)/2. This bound is tight and confirms a conjecture by Zhang…

Generalizing well-known results of Erd\H{o}s and Lov\'asz, we show that every graph $G$ contains a spanning $k$-partite subgraph $H$ with $\lambda{}(H)\geq \lceil{}\frac{k-1}{k}\lambda{}(G)\rceil$, where $\lambda{}(G)$ is the…

A graph is $k$-degenerate if every subgraph has minimum degree at most $k$. We provide lower bounds on the size of a maximum induced 2-degenerate subgraph in a triangle-free planar graph. We denote the size of a maximum induced 2-degenerate…

For graphs $G$ and $H$, an $H$-coloring of $G$ is a map from the vertices of $G$ to the vertices of $H$ that preserves edge adjacency. We consider the following extremal enumerative question: for a given $H$, which connected $n$-vertex…

We construct a family of maximal linklessly embeddable graphs on $n$ vertices and $3n-5$ edges for all $n\ge 10$, and another family on $n$ vertices and $m< \frac{25n}{12}-\frac{1}{4}$ edges for all $n\ge 13$. The latter significantly…

In 1965, Vizing proved that every planar graph $G$ with maximum degree $\Delta\geq 8$ is edge $\Delta$-colorable. It is also proved that every planar graph $G$ with maximum degree $\Delta=7$ is edge $\Delta$-colorable by Sanders and Zhao,…

Tuza's Conjecture states that if a graph $G$ does not contain more than $k$ edge-disjoint triangles, then some set of at most $2k$ edges meets all triangles of $G$. We prove Tuza's Conjecture for all graphs $G$ having no subgraph with…

A graph $ G $ is minimally $ t $-tough if the toughness of $ G $ is $ t $ and deletion of any edge from $ G $ decreases its toughness. Katona et al. conjectured that the minimum degree of any minimally $ t $-tough graph is $ \lceil 2t\rceil…

An edge in a $k$-connected graph $G$ is called {\em $k$-contractible} if the graph $G/e$ obtained from $G$ by contracting $e$ is $k$-connected. Generalizing earlier results on $3$-contractible edges in spanning trees of $3$-connected…

Halin showed that every edge minimal, k-vertex connected graph has a vertex of degree k. In this note, we prove the analogue to Halin's theorem for edge-minimal, k-edge-connected graphs. We show there are two vertices of degree k in every…

We prove that every $n$-vertex $K_t$-minor-free graph $G$ of maximum degree $\Delta$ has a set $F$ of $O(t^2(\log t)^{1/4}\sqrt{\Delta n})$ edges such that every component of $G - F$ has at most $n/2$ vertices. This is best possible up to…

We prove, that every connected graph with $s$ vertices of degree 3 and $t$ vertices of degree at least~4 has a spanning tree with at least ${2\over 5}t +{1\over 5}s+\alpha$ leaves, where $\alpha \ge {8\over 5}$. Moreover, $\alpha \ge 2$ for…

Erd\H{o}s posed the problem of finding conditions on a graph $G$ that imply the largest number of edges in a triangle-free subgraph is equal to the largest number of edges in a bipartite subgraph. We generalize this problem to general…

Tutte showed that $4$-connected planar graphs are Hamiltonian, but it is well known that $3$-connected planar graphs need not be Hamiltonian. We show that $K_{2,5}$-minor-free $3$-connected planar graphs are Hamiltonian. This does not…

We propose the following conjecture: For every fixed $\alpha\in [0,\frac 13)$, each graph of minimum degree at least $(1+\alpha)\frac k2$ and maximum degree at least $2(1-\alpha)k$ contains each tree with $k$ edges as a subgraph. Our main…

An $\textit{identifying code}$ of a closed-twin-free graph $G$ is a set $S$ of vertices of $G$ such that any two vertices in $G$ have a distinct intersection between their closed neighborhood and $S$. It was conjectured that there exists a…

Minimal prime graphs are connected graphs on at least two vertices whose complements satisfy the following conditions: triangle-freeness, 3-colorability, and edge-maximality with respect to the latter two properties. These graphs are prime…

Kriesel conjectured that every minimally $1$-tough graph has a vertex with degree precisely $2$. Katona and Varga (2018) proposed a generalized version of this conjecture which says that every minimally $t$-tough graph has a vertex with…

The concept of generalized $k$-connectivity $\kappa_{k}(G)$ of a graph $G$ was introduced by Chartrand et al. in recent years. In our early paper, extremal theory for this graph parameter was started. We determined the minimal number of…