Paw-Type Characterization of Hourglass-Free Hamilton-Connected Graphs - Panpan Wang; Liming Xiong (Axioms, 2023)
This paper introduces the forbidden subgraph conditions for Hamilton-connected graphs. If the degree sequence of the graph is <semantics>(4,2,2,2,2)</semantics> and it is connected, then it is called hourglass<semantics>Γ0</semantics>. For integers <semantics>i≥1</semantics>, the graph <semantics>Zi</semantics> is paw, which is obtained by attaching one of the vertices of the triangle to one of the end vertices of a path with a number of edges i. We show that every graph G is Hamilton-connected if G is a <semantics>Γ0</semantics>-free, <semantics>K1,3</semantics>-free, <semantics>Z14</semantics>-free, and a 3-connected graph. Moreover, we give an example to show the sharpness of a paw-type forbidden subgraph in a 3-connected, Hamilton-connected graph. Our focus on the Hamilton-connected problem can be applied to data center networks (DCNs). In the future, we will remove the forbidden subgraph families from our conclusions when building the network to obtain the optimal communication cost. Our result extends the result of Ryjáček and Vrána (Discrete Mathematics 344: 112350, 2021).