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We survey sufficient degree conditions, for a variety of graph properties, that are best possible in the same sense that Chvatal's well-known degree condition for hamiltonicity is best possible., Comment: 25 pages

We consider sufficient conditions for a degree sequence $\pi$ to be forcibly $k$-factor graphical. We note that previous work on degrees and factors has focused primarily on finding conditions for a degree sequence to be potentially $k$-factor graphical. We first give a theorem for $\pi$ to be forcibly 1-factor graphical and, more generally, forcibly graphical with deficiency at most $\beta\ge0$. These theorems are equal in strength to Chv\'atal's well-known hamiltonian theorem, i.e., the best monotone degree condition for hamiltonicity. We then give an equally strong theorem for $\pi$ to be forcibly 2-factor graphical. Unfortunately, the number of nonredundant conditions that must be checked increases significantly in moving from $k=1$ to $k=2$, and we conjecture that the number of nonredundant conditions in a best monotone theorem for a $k$-factor will increase superpolynomially in $k$. This suggests the desirability of finding a theorem for $\pi$ to be forcibly $k$-factor graphical whose algorithmic complexity grows more slowly. In the final section, we present such a theorem for any $k\ge2$, based on Tutte's well-known factor theorem. While this theorem is not best monotone, we show that it is nevertheless tight in a precise way, and give examples illustrating this tightness., Comment: 19 pages

We study theorems giving sufficient conditions on the vertex degrees of a graph $G$ to guarantee $G$ is $t$-tough. We first give a best monotone theorem when $t\ge1$, but then show that for any integer $k\ge1$, a best monotone theorem for $t=\frac1k\le 1$ requires at least $f(k)\cdot|V(G)|$ nonredundant conditions, where $f(k)$ grows superpolynomially as $k\rightarrow\infty$. When $t<1$, we give an additional, simple theorem for $G$ to be $t$-tough, in terms of its vertex degrees., Comment: 13 pages