An Etude on Recursion Relations and Triangulations

Following~\cite{Arkani-Hamed:2017thz}, we derive a recursion relation by applying a one-parameter deformation of kinematic variables for tree-level scattering amplitudes in bi-adjoint $\phi^3$ theory. The recursion relies on properties of the amplitude that can be made manifest in the underlying kinematic associahedron, and it provides triangulations for the latter. Furthermore, we solve the recursion relation and present all-multiplicity results for the amplitude: by reformulating the associahedron in terms of its vertices, it is given explicitly as a sum of "volume" of simplicies for any triangulation, which is an analogy of BCFW representation/triangulation of amplituhedron for ${\cal N}=4$ SYM.Comment: 26 pages, 3 figure

Factor-Critical Property in 3-Dominating-Critical Graphs

A vertex subset $S$ of a graph $G$ is a dominating set if every vertex of $G$ either belongs to $S$ or is adjacent to a vertex of $S$. The cardinality of a smallest dominating set is called the dominating number of $G$ and is denoted by $\gamma(G)$. A graph $G$ is said to be $\gamma$- vertex-critical if $\gamma(G-v)< \gamma(G)$, for every vertex $v$ in $G$. Let $G$ be a 2-connected $K_{1,5}$-free 3-vertex-critical graph. For any vertex $v \in V(G)$, we show that $G-v$ has a perfect matching (except two graphs), which is a conjecture posed by Ananchuen and Plummer.Comment: 8 page