Unified Spectral Hamiltonian Results of Balanced Bipartite Graphs and Complementary Graphs

Balanced coloring of bipartite graphs

Given a bipartite graph G(U ∪ V, E) with n vertices on each side, an independent set I ∈ G such that |U ⋂ I| = |V ⋂ I| is called a balanced bipartite independent set. A balanced coloring of G is a coloring of the vertices of G such that each color class induces a balanced bipartite independent set in G. If graph G has a balanced coloring we call it colorable. The coloring number χB(G) is the mi...

Bipartite Kneser Graphs are Hamiltonian

The Kneser graph K(n, k) has as vertices all k-element subsets of [n] := {1, 2, . . . , n} and an edge between any two vertices (=sets) that are disjoint. The bipartite Kneser graph H(n, k) has as vertices all k-element and (n−k)-element subsets of [n] and an edge between any two vertices where one is a subset of the other. It has long been conjectured that all connected Kneser graphs and bipar...

Spectral radius of bipartite graphs

Balanced cycles and holes in bipartite graphs

Bruce Reed asks the following question: Can we determine whether a bipartite graph contains a chordless cycle whose length is a multiple of 4? We show that the two following more general questions are equivalent and we provide an answer. Given a bipartite graph G where each edge is assigned a weight + 1 or I, l determine whether G contains a cycle whose weight is a multiple of 4, l determine wh...

Balanced list edge-colourings of bipartite graphs

Galvin solved the Dinitz conjecture by proving that bipartite graphs are ∆edge-choosable. We improve Galvin’s method and deduce from any colouring of the edges of bipartite graph G some further list edge-colouring properties of G. In particular, for bipartite graphs, it follows from the existence of balanced bipartite edge-colourings that balanced list edge-colourings exist as well. While the k...