We show that a graph of girth greater than 6 log k + 3 and minimum degree at least 3 has a minor of minimum degree greater than k. This is best possible up to a factor of at most 9/4. As a corollary, every graph of girth at least 6 log r + 3 log log r + c and minimum degree at least 3 has a Kr minor.

We give here new upper bounds on the size of a smallest feedback vertex set in planar graphs with high girth. In particular, we prove that a planar graph with girth g and size m has a feedback vertex set of size at most 4m 3g , improving the trivial bound of 2m g . We also prove that every 2-connected graph with maximum degree 3 and order n has a feedback vertex set of size at most n+2 3 .

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...

A graph G is (d1, . . . , dl)-colorable if the vertex set of G can be partitioned into subsets V1, . . . , Vl such that the graph G[Vi] induced by the vertices of Vi has maximum degree at most di for all 1 6 i 6 l. In this paper, we focus on complexity aspects of such colorings when l = 2, 3. More precisely, we prove that, for any fixed integers k, j, g with (k, j) 6= (0, 0) and g > 3, either e...

Let G be an r-regular graph of order n and independence number α(G). We show that if G has odd girth 2k + 3 then α(G) ≥ n1−1/kr1/k . We also prove similar results for graphs which are not regular. Using these results we improve on the lower bound of Monien and Speckenmeyer, for the independence number of a graph of order n and odd girth 2k + 3. AMS Subject Classification. 05C15 §

The girth of a graph G is the length of a shortest cycle in G. Dobson (1994, Ph.D. dissertation, Louisiana State University, Baton Rouge, LA) conjectured that every graph G with girth at least 2t+1 and minimum degree at least k t contains every tree T with k edges whose maximum degree does not exceed the minimum degree of G. The conjecture has been proved for t 3. In this paper, we prove Dobson...

We prove that a connected graph of diameter at least 4 and of girth 7 or more (in particular, a tree) can be exactly reconstructed from metric balls of radius 2 of all its vertices. On the other hand, there exist graphs of diameter 3 and of girth 6 which are not reconstructible. This new graph theory problem is motivated by reconstruction of chemical compounds. © 2007 Published by Elsevier B.V.

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...

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...

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...