Let $G$ be a connected general graph. For any vertex $v\in V(G)$ and any function $f: V(G)\to\mathbb{Z}^+$, we introduce a set $J_f^*(v)$ consisting of the integer $f(v)$ and all odd integers less than $f(v)$, including all negative odd integers. In this paper, we shows that the graph $G$ satisfies the general Tutte-type condition $o(G-S)\le \sum_{v\in S}f(v)$ for any nonempty set $S\subset V(G)$ if and only if either $G$ has a colored $J_f^*$-factor for any 2-end-coloring, or $G$ is of odd order and is $J_f^*$-critical for any 2-end-coloring. This characterization solves a problem posed by Akiyama and Kano, as well as a problem of Cui and Kano's.

We formulate conditions on a set of log-concave sequences, under which any linear combination of those sequences is log-concave, and further, of conditions under which linear combinations of log-concave sequences that have been transformed by convolution are log-concave. These conditions involve relations on sequences called synchronicity and ratio-dominance, and a characterization of some bivariate sequences as lexicographic. We are motivated by the 25-year-old conjecture that the genus distribution of every graph is log-concave. Although calculating genus distributions is NP-hard, they have been calculated explicitly for many graphs of tractable size, and the three conditions have been observed to occur in the partitioned genus distributions of all such graphs. They are used here to prove the log-concavity of the genus distributions of graphs constructed by iterative amalgamation of double-rooted graph fragments whose genus distributions adhere to these conditions, even though it is known that the genus polynomials of some such graphs have imaginary roots. A blend of topological and combinatorial arguments demonstrates that log-concavity is preserved through the iterations.

Let $G$ be a graph and $H\colon V(G)\to 2^\mathbb{N}$ a set function associated with $G$. A spanning subgraph $F$ of $G$ is called an $H$-factor if the degree of any vertex $v$ in $F$ belongs to the set $H(v)$. This paper contains two results on the existence of $H$-factors in regular graphs. First, we construct an $r$-regular graph without some given $H^*$-factor. In particular, this gives a negative answer to a problem recently posed by Akbari and Kano. Second, by using Lovász's characterization theorem on the existence of $(g, f)$-factors, we find a sharp condition for the existence of general $H$-factors in $\{r, r+1\}$-graphs in terms of the maximum and minimum of $H$. This result reduces to Thomassen's theorem for the case that $H(v)$ consists of the same two consecutive integers for all vertices $v$ and to Tutte's theorem if the graph is regular in addition.