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ABSTRACT A subgraph of a graph with maximum degree is ‐overfullif . Clearly, if contains a ‐overfull subgraph, then its chromatic index is . However, the converse is not true, as demonstrated by the Petersen graph. Nevertheless, three families of graphs are conjectured to satisfy the converse statement: (1) graphs with (the Overfull Conjecture of Chetwynd and Hilton), (2) planar graphs (Seymour's Exact Conjecture), and (3) graphs whose subgraph induced on the set of maximum degree vertices is the union of vertex‐disjoint cycles (the Core Conjecture of Hilton and Zhao). Over the past decades, these conjectures have been central to the study of edge coloring in simple graphs. Progress had been slow until recently, when the Core Conjecture was confirmed by the authors in 2024. This breakthrough was achieved by extending Vizing's classical fan technique to two larger families of trees: the pseudo‐multifan and the lollipop. This paper investigates the properties of these two structures, forming part of the theoretical foundation used to prove the Core Conjecture. We anticipate that these developments will provide insights into verifying the Overfull Conjecture for graphs where the subgraph induced by maximum‐degree vertices has relatively small maximum degree. 

Let $$G$$ be a $$t$$-tough graph on $$n\ge 3$$ vertices for some $t>0$. It was shown by Bauer et al. in 1995 that if the minimum degree of $$G$$ is greater than $$\frac{n}{t+1}-1$$, then $$G$$ is hamiltonian. In terms of Ore-type hamiltonicity conditions, the problem was only studied when $$t$$ is between 1 and 2, and recently the second author proved a general result. The result states that if the degree sum of any two nonadjacent vertices of $$G$$ is greater than $$\frac{2n}{t+1}+t-2$$, then $$G$$ is hamiltonian. It was conjectured in the same paper that the $+t$ in the bound $$\frac{2n}{t+1}+t-2$$ can be removed. Here we confirm the conjecture. The result generalizes the result by Bauer, Broersma, van den Heuvel, and Veldman. Furthermore, we characterize all $$t$$-tough graphs $$G$$ on $$n\ge 3$$ vertices for which $$\sigma_2(G) = \frac{2n}{t+1}-2$$ but $$G$$ is non-hamiltonian. 

Abstract A spanning tree of a graph with no vertex of degree 2 is called a homeomorphically irreducible spanning tree (HIST) of the graph. In 1990, Albertson, Berman, Hutchinson, and Thomassen conjectured that every twin‐free graph with diameter 2 contains a HIST. Recently, Ando disproved this conjecture and characterized twin‐free graphs with diameter 2 that do contain a HIST. In this paper, we give a complete characterization of all graphs of diameter 2 that contain a HIST. This characterization gives alternative proofs for several known results. 

Abstract Let be a simple graph with maximum degree . A subgraph of is overfull if . Chetwynd and Hilton in 1986 conjectured that a graph with has chromatic index if and only if contains no overfull subgraph. Let , be sufficiently large, and be graph on vertices with minimum degree at least . It was shown that the conjecture holds for if is even. In this paper, the same result is proved if is odd. As far as we know, this is the first result on the Overfull Conjecture for graphs of odd order and with a minimum degree constraint.