Let G be a connected general graph. Let f : V (G) → Z + be a function. We show that G satisfies the Tutte-type condition o(G − S) ≤ f (S) for all vertex subsets S, if and only if it contains a colored J * f-factor for any 2-end-coloring, where J * f (v) is the union of all odd integers smaller than f (v) and the integer f (v) itself. This is a generalization of the (1, f)-odd factor characterization theorem, and answers a problem of Cui and Kano. We also derive an analogous characterization for graphs of odd orders, which addresses a problem of Akiyama and Kano.

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Tittmann, Averbouch and Makowsky [P. Tittmann, I. Averbouch, J.A. Makowsky, The enumeration of vertex induced subgraphs with respect to the number of components, European Journal of Combinatorics, 32 (2011) 954-974], introduced the subgraph component polynomial Q(G; x, y) which counts the number of connected components in vertex induced subgraphs. It contains much of the underlying graph's structural information, such as the order, the size, the independence number. We show that there are several other graph invariants, like the connectivity, the number of cycles of length four in a regular bipartite graph, are determined by the subgraph component polynomial. We prove that several well-known families of graphs are determined by the polynomial Q(G; x, y). Moreover, we study the distinguishing power and find some simple graphs which are not distinguished by the subgraph component polynomial but distinguished by the character polynomial, the matching polynomial or the Tutte polynomial. These are answers to three problems proposed by Tittmann

The Tutte polynomial of a graph, or equivalently the q-state Potts model partition function, is a two-variable polynomial graph invariant of considerable importance in both combinatorics and statistical physics. The computation of this invariant for a graph is, in general, NP-hard. The aim of this paper is to compute the Tutte polynomial of the Apollonian network. Based on the well-known duality property of the Tutte polynomial, we extend the subgraph-decomposition method. In particular, we do not calculate the Tutte polynomial of the Apollonian network directly, instead we calculate the Tutte polynomial of the Apollonian dual graph. By using the close relation between the Apollonian dual graph and the Hanoi graph, we express the Tutte polynomial of the Apollonian dual graph in terms of that of the Hanoi graph. As an application, we also give the number of spanning trees of the Apollonian network.