On the least size of a graph with a given degree set

The degree set of a finite simple graph G is the set of distinct degrees of vertices of G. A theorem of Kapoor et al. [Degree sets for graphs, Fund. Math. 95 (1977) 189-194] asserts that the least order of a graph with a given degree set D is 1+max(D). We look at the analogous problem concerning the least size of a graph with a given degree set D. We determine the least size for the sets D when (i) |D|?3; (ii) D={1,2,…,n}; and (iii) every element in D is at least |D|. In addition, we give sharp upper and lower bounds in all cases.     

On diameter and inverse degree of a graph

The inverse degree r(G) of a finite graph G=(V,E) is defined as , where is the degree of vertex v. We establish inequalities concerning the sum of the diameter and the inverse degree of a graph which for the most part are tight. We also find upper bounds on the diameter of a graph in terms of its inverse degree for several important classes of graphs. For these classes, our results improve bounds by Erd?s et al. (1988) [5], and by Dankelmann et al. (2008) [4].     

Large bipartite graphs with given degree and diameter

We give constructions of bipartite graphs with maximum Δ, diameter D on B vertices, such that for every D ≥ 2 the lim inf_Δ→∞B. Δ^1-D = b_D > 0. We also improve similar results on ordinary graphs, for example, we prove that lim_Δ→∞N · Δ^?D = 1 if D is 3 or 5. This is a partial answer to a problem of Bollobás.     

Some large graphs with given degree and diameter

This paper considers the (Δ, D) problem: to maximize the order of graphs with given maximum degree Δ and diameter D, of importance for its implications in the design of interconnection networks. Two cubic graphs of diameters 5 and 8 and orders 70 and 286, respectively, and a graph of degree 5, diameter 3 and order 66 are presented, which improve the previously known orders for these values of Δ and D. By its construction, these graphs have a large automorphism group.     

Every connected regular graph of even degree is a Schreier coset graph

Using Petersen's theorem, that every regular graph of even degree is 2-factorable, it is proved that every connected regular graph of even degree is isomorphic to a Schreier coset graph. The method used is a special application of the permutation voltage graph construction developed by the author and Tucker. This work is related to graph imbedding theory, because a Schreier coset graph is a covering space of a bouquet of circles.     

Vertices of given degree in a random graph

This paper concerns the degree sequence d₁ ≥ d₂ ≥ … ≥ d_n of a randomly labeled graph of order n in which the probability of an edge is p(n) ≦ 1/2. Among other results the following questions are answered. What are the values of p(n) for which d₁, the maximum degree, is the same for almost every graph? For what values of p(n) is it true that d₂ > d₂ for almost every graph, that is, there is a unique vertex of maximum degree? The answers are (essentially) p(n) = o(logn/n/n) and p(n)n/logn → ∞. Also included is a detailed study of the distribution of degrees when 0 < lim n p(n)/log n ≦ lim n p(n)/log n < ∞.     

Large graphs with given degree and diameter. II

The following problem arises in the study of interconnection networks: find graphs of given maximum degree and diameter having the maximum number of vertices. Constructions based on a new product of graphs, which enable us to construct graphs of given maximum degree and diameter, having a great number of vertices from smaller ones are given; therefore the best values known before are improved considerably.     

Large bipartite Cayley graphs of given degree and diameter

Let BC_d,k be the largest possible number of vertices in a bipartite Cayley graph of degree d and diameter k. We show that BC_d,k≥2(k−1)((d−4)/3)^k−1 for any d≥6 and any even k≥4, and BC_d,k≥(k−1)((d−2)/3)^k−1 for d≥6 and k≥7 such that k≡3 (mod 4).     

Some properties of planar polynomial systems of even degree

Summary We study autonomous systems in the plane of polynomial type. We obtain conditions for the existence of unbounded trajectories of such systems. As a consequence we prove that it does not exist a planar polynomial system of even degree with a global center.     

On the number of vertices of given degree in a random graph

This note can be treated as a supplement to a paper written by Bollobas which was devoted to the vertices of a given degree in a random graph. We determine some values of the edge probability p for which the number of vertices of a given degree of a random graph G ∈ ??(n, p) asymptotically has a normal distribution.     

Splitting a planar graph of girth 5 into two forests with trees of small diameter

In 1985, Mihok and recently Axenovich, Ueckerdt, and Weiner asked about the minimum integer $g^1>3$ such that every planar graph with girth at least $g^1$ admits a 2-colouring of its vertices where the length of every monochromatic path is bounded from above by a constant. By results of Glebov and Zambalaeva and of Axenovich et al., it follows that $5\le g^1\le 6$. In this paper we establish that $g^1=5$. Moreover, we prove that every planar graph of girth at least 5 admits a 2-colouring of its vertices such that every monochromatic component is a tree of diameter at most 6. We also present the list version of our result.     

On the largest tree of given maximum degree in a connected graph

We prove that every connected graph G contains a tree T of maximum degree at most k that either spans G or has order at least kδ(G) + 1, where δ(G) is the minimum degree of G. This generalizes and unifies earlier results of Bermond [1] and Win [7]. We also show that the square of a connected graph contains a spanning tree of maximum degree at most three.     

Improved bound on the oriented diameter of graphs with given minimum degree

In 2015, Bau and Dankelmann showed that every bridgeless graph $G$ of order $n$ and minimum degree $\delta$ has an orientation of diameter at most $11\frac{n}{\delta +1}+9$. As they were convinced that this bound is not best possible, they posed the problem of improving it.In this paper, we prove that such a graph $G$ has an orientation of diameter less than $7\frac{n}{\delta +1}$ and give a polynomial-time algorithm to construct one.