Related papers: Complete subgraphs in a multipartite graph

A classical result by Hajnal and Szemer\'edi from 1970 determines the minimal degree conditions necessary to guarantee for a graph to contain a $K_r$-factor. Namely, any graph on $n$ vertices, with minimum degree $\delta(G) \ge…

Given a set $X$ and an integer $t$, let $\mathcal{F}$ be a family of $k$-subsets of $X$. The Kruskal--Katona theorem states that if $|\mathcal{F}|\geq \binom{t}{k}$, then $|\partial_\ell\mathcal{F}|\geq\binom{t}{\ell}$. The minimum degree…

A perfect $K_t$-matching in a graph $G$ is a spanning subgraph consisting of vertex disjoint copies of $K_t$. A classic theorem of Hajnal and Szemer\'edi states that if $G$ is a graph of order $n$ with minimum degree $\delta(G) \ge…

A $(\delta\geq k_1,\delta\geq k_2)$-partition of a graph $G$ is a vertex-partition $(V_1,V_2)$ of $G$ satisfying that $\delta(G[V_i])\geq k_i$ for $i=1,2$. We determine, for all positive integers $k_1,k_2$, the complexity of deciding…

Let $G$ be a graph of maximum degree $\Delta$ and $k$ be an integer. The $k$-recolouring graph of $G$ is the graph whose vertices are $k$-colourings of $G$ and where two $k$-colourings are adjacent if they differ at exactly one vertex. It…

A vertex-coloring of a hypergraph is conflict-free, if each edge contains a vertex whose color is not repeated on any other vertex of that edge. Let $f(r, \Delta)$ be the smallest integer $k$ such that each $r$-uniform hypergraph of maximum…

For graphs $G_0$, $G_1$ and $G_2$, write $G_0\longmapsto(G_1, G_2)$ if each red-blue-edge-coloring of $G_0$ yields a red $G_1$ or a blue $G_2$. The Ramsey number $r(G_1, G_2)$ is the minimum number $n$ such that the complete graph…

Strengthening Hadwiger's conjecture, Gerards and Seymour conjectured in 1995 that every graph with no odd $K_t$-minor is properly $(t-1)$-colorable, this is known as the Odd Hadwiger's conjecture. We prove a relaxation of the above…

We prove that any graph $G$ of minimum degree greater than $2k^2-1$ has a $(k+1)$-connected induced subgraph $H$ such that the number of vertices of $H$ that have neighbors outside of $H$ is at most $2k^2-1$. This generalizes a classical…

An $\mathcal{F}$-saturated $r$-graph is a maximal $r$-graph not containing any member of $\mathcal{F}$ as a subgraph. Let $\mathcal{K}_{\ell + 1}^{r}$ be the collection of all $r$-graphs $F$ with at most $\binom{\ell+1}{2}$ edges such that…

Let $G$ be a $t$-tough graph on $n\ge 3$ vertices for some $t>0$. It was shown by Bauer et al. in 1995 that if the minimum degree of $G$ is greater than $\frac{n}{t+1}-1$, then $G$ is hamiltonian. In terms of Ore-type hamiltonicity…

We prove that for all $k \ge 3$ and any integers $\Delta, n$ with $n \ge 2^\Delta,$ there exists a $k$-graph on $n$ vertices with maximum degree at most $\Delta$ such that $r(H)\geq\tw_{k-1}(c_k \Delta) \cdot n$ for some constant $c_k > 0$,…

Let $i_t(G)$ be the number of independent sets of size $t$ in a graph $G$. Engbers and Galvin asked how large $i_t(G)$ could be in graphs with minimum degree at least $\delta$. They further conjectured that when $n\geq 2\delta$ and $t\geq…

Let $G=(V,E)$ be a simple graph of maximum degree $\Delta$. The edges of $G$ can be colored with at most $\Delta +1$ colors by Vizing's theorem. We study lower bounds on the size of subgraphs of $G$ that can be colored with $\Delta$ colors.…

We prove that $s_r(K_{k+1}) = O(k^3 r^3 \log^3 k)$, where $s_r(K_k)$ is the Ramsey parameter introduced by Burr, Erd\H{o}s and Lov\'{a}sz in 1976, which is defined as the smallest minimum degree of a graph $G$ such that any $r$-colouring of…

Fix a color-critical graph $H$ with $\chi(H)=r+1\geq 3$. Simonovits' chromatic critical edge theorem and Nikiforov's spectral chromatic critical edge theorem imply that $T_{n,r}$ is the extremal graph with the maximum size and the maximum…

Alon, Krivelevich, and Sudakov conjectured in 1999 that for every finite graph $F$, there exists a quantity $c(F)$ such that $\chi(G) \leq (c(F) + o(1)) \Delta / \log\Delta$ whenever $G$ is an $F$-free graph of maximum degree $\Delta$. The…

Let s<t be two fixed positive integers. We study what are the minimum degree conditions for a bipartite graph G, with both color classes of size n=k(s+t), which ensure that G has a K_{s,t}-factor. Exact result for large n is given. Our…

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…

The Total Colouring Conjecture suggests that $\Delta+3$ colours ought to suffice in order to provide a proper total colouring of every graph $G$ with maximum degree $\Delta$. Thus far this has been confirmed up to an additive constant…