Related papers: A note on the threshold numbers of cycles

A graph $G=(V,E)$ is a pairwise compatibility graph (PCG) if there exists an edge-weighted tree $T$ and two non-negative real numbers $d_{min}$ and $d_{max}$, $d_{min} \leq d_{max}$, such that each node $u \in V$ is uniquely associated to a…

The Ramsey number $r(H)$ of a graph $H$ is the minimum integer $n$ such that any two-coloring of the edges of the complete graph $K_n$ contains a monochromatic copy of $H$. While this definition only asks for a single monochromatic copy of…

Let $G$ be a graph of radius $r$ and diameter $d$ with $d\leq 2r-2$. We give a new proof that $G$ contains a cycle of length at least $4r-2d$, i.e. for its circumference it holds $c(G)\geq 4r-2d$.

Threshold graphs are a class of graphs that have many equivalent definitions and have applications in integer programming and set packing problems. A graph is said to have a threshold cover of size $k$ if its edges can be covered using $k$…

Given a graph $G$ and a coloring of its edges, a subgraph of $G$ is called rainbow if its edges have distinct colors. The rainbow girth of an edge coloring of G is the minimum length of a rainbow cycle in G. A generalization of the famous…

Erd\"os conjectured that if $G$ is a triangle free graph of chromatic number at least $k\geq 3$, then it contains an odd cycle of length at least $k^{2-o(1)}$ \cite{sudakovverstraete, verstraete}. Nothing better than a linear bound…

Let $r$ be a positive integer and $G$ be a graph. The list $r$-hued chromatic number of $G$, denoted by $\chi_{L,r}(G)$, is the smallest integer $k$, such that for each $k$-list $L$ of $G$, $G$ has an $(L,r)$-coloring. It is proved in…

We show that $p=\sqrt{\frac{e}{n}}$ is a sharp threshold for the random graph $G_{n,p}$ to contain the square of a Hamilton cycle. This improves the previous results of K\"uhn and Osthus and also Nenadov and \v{S}kori\'c.

In a graph whose vertices are assigned integer ranks, a path is well-ranked if the endpoints have distinct ranks or some interior point has a higher rank than the endpoints. A ranking is an assignment of ranks such that all nontrivial paths…

In this short note we determine the maximum number, over all $n$-vertex graphs $G$, of orientations of $G$ containing no strongly connected cycle $C_{2k+1}$. This answers a part of a recent question of Araujo, Botler and Mota.

A graph $ G $ is said to be $ (H;k) $-vertex stable if $ G $ contains a~subgraph isomorphic to $ H $ even after removing any $ k $ of its vertices alongside with their incident edges. We will denote by $ \text{stab}(H;k) $ the minimum size…

For a positive integer $k$, the $k$-cut complex of a graph $G$ is the simplicial complex whose facets are the $(|V(G)|-k)$-subsets $\sigma$ of the vertex set $V(G)$ of $G$ such that the induced subgraph of $G$ on $V(G) \setminus \sigma$ is…

We consider the size of the smallest set of vertices required to intersect every longest path in a chordal graph. Such sets are known as longest path transversals. We show that if $\omega(G)$ is the clique number of a chordal graph $G$,…

We revisit results obtained in [F. Harary, U. Peled, Hamiltonian threshold graphs, Discrete Appl.~Math., 16 (1987), 11--15], where several necessary and necessary and sufficient conditions for a connected threshold graph to be Hamiltonian…

Let $\Gamma(n,k)$ be the set of $2$-connected $n$-vertex graphs containing an edge that is not on any cycle of length at least $k+1.$ Let $g_s(n,k)$ denote the maximum number of $s$-cliques in a graph in $\Gamma(n,k).$ Recently, Ji and Ye…

The \textit{minimum positive co-degree} of a non-empty $r$-graph $H$, denoted $\delta_{r-1}^+(H)$, is the largest integer $k$ such that if a set $S \subset V(H)$ of size $r-1$ is contained in at least one $r$-edge of $H$, then $S$ is…

It is known that $\Theta(\log n)$ chords must be added to an $n$-cycle to produce a pancyclic graph; for vertex pancyclicity, where every vertex belongs to a cycle of every length, $\Theta(n)$ chords are required. A possibly `intermediate'…

For a graph $G$ and $S\subset V(G)$, if $G - S$ is acyclic, then $S$ is said to be a decycling set of $G$. The size of a smallest decycling set of $G$ is called the decycling number of $G$. The purpose of this paper is a comprehensive…

A graph $G$ is $k$-critical (list $k$-critical, DP $k$-critical) if $\chi(G)= k$ ($\chi_\ell(G)= k$, $\chi_\mathrm{DP}(G)= k$) and for every proper subgraph $G'$ of $G$, $\chi(G')<k$ ($\chi_\ell(G')< k$, $\chi_\mathrm{DP}(G')<k$). Let $f(n,…

For integers $k>0$ and $0<r \leq \Delta$ (where $r \leq k$), a conditional $(k,r)$-coloring of a graph $G$ is a proper $k$-coloring of the vertices of $G$ such that every vertex $v$ of degree $d(v)$ in $G$ is adjacent to vertices with at…