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The clique removal lemma says that for every $r geq 3$ and $varepsilon>0$, there exists some $delta>0$ so that every $n$-vertex graph $G$ with fewer than $delta n^r$ copies of $K_r$ can be made $K_r$-free by removing at most $varepsilon n^2$ edges. The dependence of $delta$ on $varepsilon$ in this result is notoriously difficult to determine: it is known that $delta^{-1}$ must be at least super-polynomial in $varepsilon^{-1}$, and that it is at most of tower type in $log varepsilon^{-1}$. We prove that if one imposes an appropriate minimum degree condition on $G$, then one can actually take $delta$ to be a linear function of $varepsilon$ in the clique removal lemma. Moreover, we determine the threshold for such a minimum degree requirement, showing that above this threshold we have linear bounds, whereas below the threshold the bounds are once again super-polynomial, as in the unrestricted removal lemma. We also investigate this question for other graphs besides cliques, and prove some general results about how minimum degree conditions affect the bounds in the graph removal lemma.
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ErdH{o}s determined the maximum size of a nonhamiltonian graph of order $n$ and minimum degree at least $k$ in 1962. Recently, Ning and Peng generalized. ErdH{o}s work and gave the maximum size $h(n,c,k)$ of graphs with prescribed order $n$, circumference $c$ and minimum degree at least $k.$ But for some triples $n,c,k,$ the maximum size is not attained by a graph of minimum degree $k.$ For example, $h(15,14,3)=77$ is attained by a unique graph of minimum degree $7,$ not $3.$ In this paper we obtain more precise information by determining the maximum size of a graph with prescribed order, circumference and minimum degree. Consequently we solve the corresponding problem for longest paths. All these results on the size of graphs have cliq
Given a simple graph $G$, denote by $Delta(G)$, $delta(G)$, and $chi(G)$ the maximum degree, the minimum degree, and the chromatic index of $G$, respectively. We say $G$ is emph{$Delta$-critical} if $chi(G)=Delta(G)+1$ and $chi(H)le Delta(G)$ for every proper subgraph $H$ of $G$; and $G$ is emph{overfull} if $|E(G)|>Delta lfloor |V(G)|/2 rfloor$. Since a maximum matching in $G$ can have size at most $lfloor |V(G)|/2 rfloor$, it follows that $chi(G) = Delta(G) +1$ if $G$ is overfull. Conversely, let $G$ be a $Delta$-critical graph. The well known overfull conjecture of Chetwynd and Hilton asserts that $G$ is overfull provided $Delta(G) > |V(G)|/3$. In this paper, we show that any $Delta$-critical graph $G$ is overfull if $Delta(G) - 7delta(G)/4ge(3|V(G)|-17)/4$.
The degree-based entropy of a graph is defined as the Shannon entropy based on the information functional that associates the vertices of the graph with the corresponding degrees. In this paper, we study extremal problems of finding the graphs attaining the minimum degree-based graph entropy among graphs and bipartite graphs with a given number of vertices and edges. We characterize the unique extremal graph achieving the minimum value among graphs with a given number of vertices and edges and present a lower bound for the degree-based entropy of bipartite graphs and characterize all the extremal graphs which achieve the lower bound. This implies the known result due to Cao et al. (2014) that the star attains the minimum value of the degree-based entropy among trees with a given number of vertices.
Let $G$ be a simple graph with maximum degree $Delta(G)$. A subgraph $H$ of $G$ is overfull if $|E(H)|>Delta(G)lfloor |V(H)|/2 rfloor$. Chetwynd and Hilton in 1985 conjectured that a graph $G$ with $Delta(G)>|V(G)|/3$ has chromatic index $Delta(G)$ if and only if $G$ contains no overfull subgraph. The 1-factorization conjecture is a special case of this overfull conjecture, which states that for even $n$, every regular $n$-vertex graph with degree at least about $n/2$ has a 1-factorization and was confirmed for large graphs in 2014. Supporting the overfull conjecture as well as generalizing the 1-factorization conjecture in an asymptotic way, in this paper, we show that for any given $0<varepsilon <1$, there exists a positive integer $n_0$ such that the following statement holds: if $G$ is a graph on $2nge n_0$ vertices with minimum degree at least $(1+varepsilon)n$, then $G$ has chromatic index $Delta(G)$ if and only if $G$ contains no overfull subgraph.
We prove that $s_r(K_k) = O(k^5 r^{5/2})$, where $s_r(K_k)$ is the Ramsey parameter introduced by Burr, ErdH{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 the edges of $G$ contains a monochromatic $K_k$, whereas no proper subgraph of $G$ has this property. The construction used in our proof relies on a group theoretic model of generalised quadrangles introduced by Kantor in 1980.
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