Nearly bipartite graphs

Nearly bipartite graphs

We prove that if a nonbipartite graph G on n vertices has minimal degree δ ≥ n 4k + 2 + ck,m where ck,m does not depend on n and n is sufficiently large, if C2s+1 ⊂ G for some k ≤ s ≤ 4k + 1 then C2s+2j+1 ⊂ G for every j = 1, ...,m. We give a structural description of all graphs on n vertices with

Nonorientable Genus of Nearly Complete Bipartite Graphs

On f -Edge Cover Coloring of Nearly Bipartite Graphs

Let G(V,E) be a graph, and let f be an integer function on V with 1 ≤ f(v) ≤ d(v) to each vertex v ∈ V . An f -edge cover coloring is an edge coloring C such that each color appears at each vertex v at least f(v) times. The f -edge cover chromatic index of G, denoted by χ ′ fc(G), is the maximum number of colors needed to f -edge cover color G. It is well known that min v∈V {bd(v)− μ(v) f(v) c ...

Graphs without short odd cycles are nearly bipartite

It is proved that for every constant ~ > 0 and every graph G on n vertices which contains no odd cycles of length smaller than ~n, G can be made bipartite by removing (15/~}ln(10/~)) vertices. This resuIt is best possible except for a constant factor. Moreover, it is shown that one can destroy all odd cycles in such a graph G alto by omitting not more thaa (200/e')0n(10/e)) z edges. For a graph...

High-Girth Graphs Avoiding a Minor are Nearly Bipartite

Let H be a xed graph. We show that any H-minor free graph of high enough girth has a circular-chromatic number arbitrarily close to two. Equivalently, such graphs have homomorphisms into a large odd circuit. In particular, graphs of high girth and of bounded genus or bounded tree width are \nearly bipartite" in this sense. For example, any planar graph of girth at least 16 admits a homomorphism...