Journal of Graph Theory

The Kneser graph K(n, k) has as its vertex set all k-subsets of an n-set and two k-subsets are adjacent if they are disjoint. The odd graph O_k is a special case of Kneser graph when n = 2k + 1. A long standing conjecture claims that O_k is hamiltonian ...

We show that for each rational number r such that 4<r⩽5 there exist infinitely many cyclically 4-edge-connected cubic graphs of chromatic index 4 and girth at least 5—that is, snarks—whose flow number equals r. This answers a question posed by Pan and ...

In 1968, Vizing conjectured that if G is a Δ-critical graph with n vertices, then α(G)≤n/2, where α(G) is the independence number of G. In this paper, we apply Vizing and Vizing-like adjacency lemmas to this problem and prove that α(G)<(((5Δ−6)n)/(8Δ−6))...

We identify the structures of 4-connected projective-planar graphs which generate their inequivalent embeddings on the projective plane, showing two series of graphs the number of whose inequivalent embeddings is held by O(n) with respect to the number ...

Map the vertices of a graph to (not necessarily distinct) points of the plane so that two adjacent vertices are mapped at least unit distance apart. The plane-width of a graph is the minimum diameter of the image of its vertex set over all such ...

A near-polygonal graph is a graph Γ which has a set 𝒞 of m-cycles for some positive integer m such that each 2-path of Γ is contained in exactly one cycle in 𝒞. If m is the girth of Γ then the graph is called polygonal. Given a polygonal graph Γ of ...

Motivated by the work of Nešetřil and Rödl on “Partitions of vertices” we are interested in obtaining some quantitative extensions of their result. In particular, given a natural number r and a graph G of order m with odd girth g, we show the existence ...