Related papers: A note on $k$-cordial $p$-uniform hypertrees

For a $k$-uniform hypergraph $F$ let $\textrm{ex}(n,F)$ be the maximum number of edges of a $k$-uniform $n$-vertex hypergraph $H$ which contains no copy of $F$. Determining or estimating $\textrm{ex}(n,F)$ is a classical and central problem…

A graph H is k-common if the number of monochromatic copies of H in a k-edge-coloring of K_n is asymptotically minimized by a random coloring. For every k, we construct a connected non-bipartite k-common graph. This resolves a problem…

We give a precise description of combed trees in terms of Kelly-Mac Lane graphs. We show that any combed tree is uniquely expressed as an allowable Kelly-Mac Lane graph of a certain shape. Conversely, we show that any such Kelly-Mac Lane…

Motivated by the study of random temporal networks, we introduce a class of random trees that we coin \emph{uniform temporal trees}. A uniform temporal tree is obtained by assigning independent uniform $[0,1]$ labels to the edges of a…

For all integers $k\geq 3$, we give an $O(n^4)$ time algorithm for the problem whose instance is a graph $G$ of girth at least $k$ together with $k$ vertices and whose question is "Does $G$ contains an induced subgraph containing the $k$…

For a finite family $\mathcal{F}$ of fixed graphs let $R_k(\mathcal{F})$ be the smallest integer $n$ for which every $k$-coloring of the edges of the complete graph $K_n$ yields a monochromatic copy of some $F\in\mathcal{F}$. We say that…

Let $G$ be a nontrivial connected graph with an edge-coloring $c:E(G)\rightarrow \{1,2,\ldots,q\},$ $q\in \mathbb{N}$, where adjacent edges may be colored the same. A tree $T$ in $G$ is a $rainbow tree$ if no two edges of $T$ receive the…

The Surjective Homomorphism problem is to test whether a given graph G called the guest graph allows a vertex-surjective homomorphism to some other given graph H called the host graph. The bijective and injective homomorphism problems can…

The transversal number $\tau(H)$ of a hypergraph $H$ is the minimum number of vertices that intersect every edge of $H$. A linear hypergraph is one in which every two distinct edges intersect in at most one vertex. A $k$-uniform hypergraph…

Mader [J. Combin. Theory Ser. B 40 (1986) 152-158] proved that every $k$-edge-connected graph $G$ with minimum degree at least $k+1$ contains a vertex $u$ such that $G-\{u\}$ is still $k$-edge-connected. In this paper, we prove that every…

A $k$-graph system $\textbf{H}=\{H_i\}_{i\in[m]}$ is a family of not necessarily distinct $k$-graphs on the same $n$-vertex set $V$ and a $k$-graph $H$ on $V$ is said to be $\textbf{H}$-transversal provided that there exists an injection…

Camarena, Cs\'{o}ka, Hubai, Lippner, and Lov\'{a}sz introduced the notion of positive graphs. This notion naturally extends to $r$-uniform hypergraphs. In the case when $r$ is odd, we prove that a hypergraph is positive if and only if its…

We show that, for every $k \ge 2$, every $k$-uniform hypergaph of degree $\Delta$ and girth at least $5$ is efficiently $(1+o(1) )(k-1) (\Delta / \ln \Delta )^{ 1/(k-1) } $-list colorable. As an application (and to the best of our…

Given a graph $G$, the Bell $k$-coloring graph $\mathcal{B}_k(G)$ has vertices given by partitions of $V(G)$ into $k$ independent sets (allowing empty parts), with two partitions adjacent if they differ only in the placement of a single…

Let $p, k, q$ be positive integers with $p-2 \geqslant k$ and let $K_{p,k}^{q}$ be the generalized pineapple graph which is obtained by joining independent set of $q$ vertices with $k$ vertices of a complete graph $K_{p}.$ In \cite{TSH2},…

In this note, we determine the maximum number of edges of a $k$-uniform hypergraph, $k\ge 3$, with a unique perfect matching. This settles a conjecture proposed by Snevily.

Given a graph $H$, a graph $G$ is $H$-free if $G$ does not contain $H$ as an induced subgraph. For a positive real number $t$, a non-complete graph $G$ is said to be $t$-tough if for every vertex cut $S$ of $G$, the ratio of $|S|$ to the…

Chung and Graham considered the problem of minimizing the number of edges in an $n$-vertex graph containing all $n$-vertex trees as a subgraph. They showed that such a graph has at least $\frac{1}{2}n \log{n}$ edges. In this note, we…

Generalizing the decomposition of a connected planar graph into a tree and a dual tree, we prove a combinatorial analog of the classic Helmholz-Hodge decomposition of a smooth vector field. Specifically, we show that for every polyhedral…

In this paper, we introduce a generalized version of an ear decomposition, called a $j$-spider decomposition, for $j$-connected star-free graphs with $j \geq 2$. Its application enables us to improve a previousely known sufficient condition…