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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
Motivated by work of ErdH{o}s, Ota determined the maximum size $g(n,k)$ of a $k$-connected nonhamiltonian graph of order $n$ in 1995. But for some pairs $n,k,$ the maximum size is not attained by a graph of connectivity $k.$ For example, $g(15,3)=77$
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 eve
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. T
The degree set of a finite simple graph $G$ is the set of distinct degrees of vertices of $G$. A theorem of Kapoor, Polimeni & Wall asserts that the least order of a graph with a given degree set $mathscr D$ is $1+max mathscr D$. Tripathi & Vijay con
An edge-coloured graph $G$ is called $properly$ $connected$ if every two vertices are connected by a proper path. The $proper$ $connection$ $number$ of a connected graph $G$, denoted by $pc(G)$, is the smallest number of colours that are needed in or