We provide proofs of the following theorems by considering the entropy of random walks. Theorem 1.(Alon, Hoory and Linial) Let G be an undirected simple graph with n vertices, girth g, minimum degree at least 2 and average degree d̄. Odd girth: If g = 2r + 1, then n ≥ 1 + d̄ r−1 ∑ i=0 (d̄− 1)i. Even girth: If g = 2r, then n ≥ 2 r−1 ∑ i=0 (d̄− 1)i. Theorem 2.(Hoory) Let G = (VL, VR, E) be a bipartit...

Recently, Borodin, Kostochka, and Yancey (On 1-improper 2-coloring of sparse graphs. Discrete Mathematics, 313(22), 2013) showed that the vertices of each planar graph of girth at least 7 can be 2-colored so that each color class induces a subgraph of a matching. We prove that any planar graph of girth at least 6 admits a vertex coloring in 2 colors such that each monochromatic component is a p...

We study the structure of graphs with high minimum degree conditions and given odd girth. For example, the classical work of Andrásfai, Erdős, and Sós implies that every n-vertex graph with odd girth 2k + 1 and minimum degree bigger than 2n 2k+1 must be bipartite. We consider graphs with a weaker condition on the minimum degree. Generalizing results of Häggkvist and of Häggkvist and Jin for the...

We study extremal problems for decomposing a connected n-vertex graph G into trees or into caterpillars. The least size of such a decomposition is the tree thickness θT(G) or caterpillar thickness θC(G). If G has girth g with g ≥ 5, then θT(G) ≤ bn/gc + 1. We conjecture that the bound holds also for g = 4 and prove it when G contains no subdivision of K2,3 with girth 4. For θC, we prove that θC...

For a finite, simple, undirected graph G and an integer d ≥ 1, a mindeg-d subgraph is a subgraph of G of minimum degree at least d. The dgirth of G, denoted gd(G), is the minimum size of a mindeg-d subgraph of G. It is a natural generalization of the usual girth, which coincides with the 2-girth. The notion of d-girth was proposed by Erdős et al. [13, 14] and Bollobás and Brightwell [7] over 20...

For 3 ≤ k ≤ 20 with k 6= 4, 8, 12, all the smallest currently known k–regular graphs of girth 5 have the same orders as the girth 5 graphs obtained by the following construction: take a (not necessarily Desarguesian) elliptic semiplane S of order n− 1 where n = k − r for some r ≥ 1; the Levi graph Γ (S) of S is an n–regular graph of girth 6; parallel classes of S induce co–cliques in Γ (S), som...

Let Γ denote a finite, connected, simple graph. For an edge e of let n(e) the number girth cycles containing e. vertex v {e1, e2, …, ek} be set edges incident to ordered such that n(e1) ≤ n(e2) … n(ek). Then (n(e1),n(e2),…,n(ek)) is called signature v. The graph said girth-biregular if it bipartite, and all its vertices belonging same bipartition have signature. with g = 2d signatures (a1,a2,…,...