The clique removal lemma says that for every r ≥ 3 $r\ge 3$ and ε > 0 $\varepsilon \gt 0$ , there exists some δ $\delta so n $n$ -vertex graph G $G$ with fewer than {n}^{r}$ copies of K ${K}_{r}$ can be made -free by removing at most 2 {n}^{2}$ edges. dependence $ on in this result is notoriously difficult to determine: it known − 1 ${\delta }^{-1}$ must least super-polynomial ${\varepsilon tow...

Let G be a connected graph with n vertices. Then the class of graphs having vertices is denoted by Gn. The subclass 5 cycles are Gn5. classification G∈Gn5 depends on number edges and sum degrees graph. Any in Gn5 contains five linearly independent at least n+3 5-cyclic must equal to twice n+4. In this paper, minimum degree distance cyclic investigated. To find some transformations T have been d...

A classical result of Erd?s, Gyárfás and Pyber states that any r-edge-coloured complete graph has a partition into O(r2log?r) monochromatic cycles. Here we determine the minimum degree threshold for this property. More precisely, show there exists constant c such on n vertices with at least n/2+c?rlog?n O(r2) We also provide constructions showing condition number cycles are essentially tight.

We develop a new framework to study minimum d $d$ -degree conditions in k $k$ -uniform hypergraphs, which guarantee the existence of tight Hamilton cycle. Our main theoretical result deals with typical absorption, path cover and connecting arguments for all at once, thus sheds light on underlying structural problems. Building this, we show that one can cycles by focusing inner structure neighbo...

Let G $G$ be a simple graph with maximum degree Δ ( ) ${\rm{\Delta }}(G)$ . A subgraph H $H$ of is overfull if ∣ E > ⌊ V ∕ 2 ⌋ $| E(H)| \gt {\rm{\Delta }}(G)\lfloor | V(H)| \unicode{x02215}2\rfloor .$ Chetwynd and Hilton in 1986 conjectured that 3 }}(G)\gt V(G)| \unicode{x02215}3$ has chromatic index only contains no subgraph. The best previous results supporting this conjecture have been obtai...