Let $G$ be an $r$-partite graph such that the edge density between any two parts is at least $\alpha$. We consider the problem of determining how large $\alpha$ must be in order to guarantee that $G$ has a Hamiltonian traversal (an $r$-cycle subgraph containing exactly one vertex from each part), and show that this critical density tends to $\frac 1 2$ as $r$ increases. This resolves a conjecture of Badakhshian, Falgas-Ravry, and Sharifzadeh. We also study the critical densities necessary to guarantee the existence of other spanning structures in traversals, particularly subgraph factors, and obtain asymptotically the critical densities for traversal $F$-factor subgraphs for several classes of graphs $F$. The proofs of our results involve the absorption method.
An old result of Tutte states that any $d$-regular graph contains a spanning subgraph in which every vertex has degree $k$ or $k+1$, for every $1\leq k\leq d$. We generalize this statement to hypergraphs, showing, for example, that every $3$-uniform $d$-regular hypergraph contains a subgraph in which all degrees are $k...
Noga Alon, P. Haxell, Aleksa Milojević et al.· 0 citations
We prove that there is an absolute constant $c>0$ such that every graph of chromatic number at least $r$ and at most $cr^3\log^2 r$ edges contains at least $\binom r3$ triangles. The proof has three ingredients. First, a sparse-core argument based on a triangle-sensitive coloring estimate of Harris extracts, from any c...
For a graph $G$, an odd induced subgraph of $G$ is an induced subgraph in which every vertex has odd degree (within the subgraph). Let $f_o(G)$ denote the maximum size of such a subgraph in $G$. Caro conjectured that there exists a positive constant $c$ such that $f_o(G)≥cn$ for any $n$-vertex graph without isolated ve...
Xin-Ru Yang, Qing-Hou Zeng· Annals of Applied Mathematic...· 0 citations
We show that almost every graph admits a partition of its vertex set into three parts such that no two adjacent vertices have the same number of neighbors in each of the three parts. Equivalently, for $G\sim G(n,1/2)$, $\chi_m(G)\le3$ with high probability, improving the previously known bound of five. Here $\chi_m(G)$...
A set $S$ of vertices of a graph $G$ is a connected mutual-visibility set if every two vertices of $S$ are joined by a shortest path whose internal vertices lie outside $S$, and the subgraph induced by $S$ is connected. We introduce the connected mutual-visibility number $\mu_c(G)$, defined as the maximum cardinality o...
A finite connected graph is rarely a Cayley graph. We measure how far it is from being one: given $G$ with $n$ vertices and $m$ edges, how few edges must be added, or added and deleted, before the result is a Cayley graph of an abelian group of order $n$ on the same vertex set? This defines two invariants, the completi...
