We prove that for graphs of order n, minimum degree δ ≥ 2 and girth g ≥ 5 the domination number γ satisfies γ ≤ ( 1 3 + 2 3g ) n. As a corollary this implies that for cubic graphs of order n and girth g ≥ 5 the domination number γ satisfies γ ≤ ( 44 135 + 82 135g ) n which improves recent results due to Kostochka and Stodolsky (An upper bound on the domination number of n-vertex connected cubic...

Consider the following problem: Does there exist a graph with large girth and large chromatic number? The girth of a graph is the length of the shortest cycle and the chromatic number is the minimum number of colors needed to color the vertices of the graph so no two adjacent vertices have the same color. These two conditions seem contradictory, intuitively to have high chromatic number we need...

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...

The objective of this study was to identify morphological measurements that best distinguish Moroccan Barcha and Atlas goat breeds. Ten measurements (body weight - BW, body length - BL, heart girth - HG, withers height - WH, rump height - WH, back length - BAL, neck length - NL, head length - HL, ear length - EL, and ear width - EW) of 876 adult animals of both sexes (547 Barcha and 329 Atlas) ...

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...