A d-angulation is a planar map with faces of degree d. We present for each integer d ≥ 3 a bijection between the class of d-angulations of girth d and a class of decorated plane trees. Each of the bijections is obtained by specializing a “master bijection” which extends an earlier construction of the first author. Bijections already existed for triangulations (d = 3) and for quadrangulations (d...

We show that every (sub)cubic n-vertex graph with sufficiently large girth has fractional chromatic number at most 2.2978 which implies that it contains an independent set of size at least 0.4352n. Our bound on the independence number is valid to random cubic graphs as well as it improves existing lower bounds on the maximum cut in cubic graphs with large girth.

We show that a planar graph with girth at least 20t−2 3 has circular chromatic number at most 2+ 1t , improving earlier results. This follows from a general result establishing homomorphisms into special targets for graphs with given girth and given maximum average degree. Other applications concern oriented chromatic number and homomorphisms into mixed graphs with colored edges.

We describe some new applications of nonabelian pq-groups to construction problems in Graph Theory. The constructions include the smallest known trivalent graph of girth 17, the smallest known regular graphs of girth five for several degrees, along with four edge colorings of complete graphs that improve lower bounds on classical Ramsey numbers.

Let q be a prime a power and k an integer such that 3 ≤ k ≤ q. In this paper we present a method using Latin squares to construct adjacency matrices of k-regular bipartite graphs of girth 8 on 2(kq − q) vertices. Some of these graphs have the smallest number of vertices among the known regular graphs with girth 8.

We prove that, the acyclic chromatic index a′(G) ≤ 6∆ for all graphs with girth at least 9. We extend the same method to obtain a bound of 4.52∆ with the girth requirement g ≥ 220. We also obtain a relationship between g and a′(G).

A star coloring of a graph is a proper vertex-coloring such that no path on four vertices is 2-colored. We prove that the vertices of every planar graph of girth 6 (respectively 7,8) can be star colored from lists of size 8 (respectively 7,6). We give an example of a planar graph of girth 5 that requires 6 colors to star color.

This works presents a formalization of the Girth-Chromatic number theorem in graph theory, stating that graphs with arbitrarily large girth and chromatic number exist. The proof uses the theory of Random Graphs to prove the existence with probabilistic arguments and is based on [1].

We study the asymptotic value of several extremal problems on graphs and hypergraphs, that arise as generalized notions of girth. Apart from being combinatorially natural questions, they are motivated by computational-theoretic applications. 1. An `-subgraph is a subgraph with ` edges per vertex, or equivalently, average degree 2`. What is the optimal upper bound S`(n, d), such that any graph o...

Reiman’s inequality for the size of bipartite graphs of girth six is generalised to girth eight. It is optimal in as far as it admits the algebraic structure of generalised quadrangles as case of equality. This enables us to obtain the optimal estimate e ∼ v for balanced bipartite graphs. We also get an optimal estimate for very unbalanced graphs.