The smallest 2-connected cubic bipartite planar nonhamiltonian graph

Nonhamiltonian 3-Connected Cubic Planar Graphs

We establish that every cyclically 4-connected cubic planar graph of order at most 40 is hamiltonian. Furthermore, this bound is determined to be sharp and we present all nonhamiltonian such graphs of order 42. In addition we list all nonhamiltonian cyclically 5-connected cubic planar graphs of order at most 52 and all nonhamiltonian 3-connected cubic planar graphs of girth 5 on at most 46 vert...

The smallest non-hamiltonian 3-connected cubic planar graphs have 38 vertices

Finite groups admitting a connected cubic integral bi-Cayley graph

A graph   is called integral if all eigenvalues of its adjacency matrix  are integers.  Given a subset $S$ of a finite group $G$, the bi-Cayley graph $BCay(G,S)$ is a graph with vertex set $Gtimes{1,2}$ and edge set ${{(x,1),(sx,2)}mid sin S, xin G}$.  In this paper, we classify all finite groups admitting a connected cubic integral bi-Cayley graph.

The Planar Algebra of a bipartite graph

Characterizing degree-sum maximal nonhamiltonian bipartite graphs

In 1963, Moon and Moser gave a bipartite analogue to Ore’s famed theorem on hamiltonian graphs. While the sharpness examples of Ore’s Theorem have been independently characterized in at least four different papers, no similar characterization exists for the Moon-Moser Theorem. In this note, we give such a characterization, consisting of one infinite family and two exceptional graphs of order ei...