On bipartite graphs of diameter 3 and defect 2

We consider bipartite graphs of degree Δ≥2, diameter D=3, and defect 2 (having 2 vertices less than the bipartite Moore bound). Such graphs are called bipartite (Δ, 3, ?2) ‐graphs. We prove the uniqueness of the known bipartite (3, 3, ?2) ‐graph and bipartite (4, 3, ?2)‐graph. We also prove several necessary conditions for the existence of bipartite (Δ, 3, ?2) ‐graphs. The most general of these conditions is that either Δ or Δ?2 must be a perfect square. Furthermore, in some cases for which the condition holds, in particular, when Δ=6 and Δ=9, we prove the non‐existence of the corresponding bipartite (Δ, 3, ?2)‐graphs, thus establishing that there are no bipartite (Δ, 3, ?2)‐graphs, for 5≤Δ≤10. © 2009 Wiley Periodicals, Inc. J Graph Theory 61: 271–288, 2009     

A family of graphs and the degree/diameter problem

We investigate a family of graphs relevant to the problem of finding large regular graphs with specified degree and diameter. Our family contains the largest known graphs for degree/diameter pairs (3, 7), (3, 8), (4, 4), (5, 3), (5, 5), (6, 3), (6, 4), (7, 3), (14, 3), and (16, 2). We also find a new bound for (3, 6) by an unrelated method. © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 118–124, 2001     

Complete characterization of almost Moore digraphs of degree three

It is well known that Moore digraphs do not exist except for trivial cases (degree 1 or diameter 1), but there are digraphs of diameter two and arbitrary degree which miss the Moore bound by one. No examples of such digraphs of diameter at least three are known, although several necessary conditions for their existence have been obtained. In this paper, we prove that digraphs of degree three and diameter k ≥ 3 which miss the Moore bound by one do not exist. © 2004 Wiley Periodicals, Inc. J Graph Theory 48: 112–126, 2005     

Graphs of order two less than the Moore bound

The problem of determining the largest order n_d,k of a graph of maximum degree at most d and diameter at most k is well known as the degree/diameter problem. It is known that n_d,k?M_d,k where M_d,k is the Moore bound. For d=4, the current best upper bound for n_4,k is M_4,k-1. In this paper we study properties of graphs of order M_d,k-2 and we give a new upper bound for n_4,k for k?3.     

An improved upper bound for the order of mixed graphs

A mixed graph $G$ can contain both (undirected) edges and arcs (directed edges). Here we derive an improved Moore-like bound for the maximum number of vertices of a mixed graph with diameter at least three. Moreover, a complete enumeration of all optimal $(1,1)$-regular mixed graphs with diameter three is presented, so proving that, in general, the proposed bound cannot be improved.     

An improved Moore bound and some new optimal families of mixed Abelian Cayley graphs

We consider the case in which mixed graphs (with both directed and undirected edges) are Cayley graphs of Abelian groups. In this case, some Moore bounds were derived for the maximum number of vertices that such graphs can attain. We first show these bounds can be improved if we know more details about the order of some elements of the generating set. Based on these improvements, we present some new families of mixed graphs. For every fixed value of the degree, these families have an asymptotically large number of vertices as the diameter increases. In some cases, the results obtained are shown to be optimal.     

All hypoenergetic graphs with maximum degree at most 3

The energy E(G) of a graph G is defined as the sum of the absolute values of its eigenvalues. A connected graph G of order n is said to be hypoenergetic if E(G)<n. All connected hypoenergetic graphs with maximum degree Δ3 have been characterized. In addition to the four (earlier known) hypoenergetic trees, we now show that complete bipartite graph K_2,3 is the only hypoenergetic cycle-containing hypoenergetic graph. By this, the validity of a conjecture by Majstorović et al. has been confirmed.     

A 4/3-approximation for TSP on cubic 3-edge-connected graphs

For a 3-edge-connected cubic graph $G=(V,E)$, we give an algorithm to construct a connected Eulerian subgraph of $2G$ using at most $?4\left|V\right|∕3?$ edges.     

The firefighter problem for graphs of maximum degree three

We show that the firefighter problem is NP-complete for trees of maximum degree three, but in P for graphs of maximum degree three if the fire breaks out at a vertex of degree at most two.     

On the existence of graphs of diameter two and defect two

In the context of the degree/diameter problem, the ‘defect’ of a graph represents the difference between the corresponding Moore bound and its order. Thus, a graph with maximum degree d and diameter two has defect two if its order is n=d²−1. Only four extremal graphs of this type, referred to as (d,2,2)-graphs, are known at present: two of degree d=3 and one of degree d=4 and 5, respectively. In this paper we prove, by using algebraic and spectral techniques, that for all values of the degree d within a certain range, (d,2,2)-graphs do not exist.The enumeration of (d,2,2)-graphs is equivalent to the search of binary symmetric matrices A fulfilling that AJ_n=dJ_n and A²+A+(1−d)I_n=J_n+B, where J_n denotes the all-one matrix and B is the adjacency matrix of a union of graph cycles. In order to get the factorization of the characteristic polynomial of A in Q[x], we consider the polynomials F_i,d(x)=f_i(x²+x+1−d), where f_i(x) denotes the minimal polynomial of the Gauss period , being ζ_i a primitive ith root of unity. We formulate a conjecture on the irreducibility of F_i,d(x) in Q[x] and we show that its proof would imply the nonexistence of (d,2,2)-graphs for any degree d>5.     

Vertex decompositions of sparse graphs into an edgeless subgraph and a subgraph of maximum degree at most k

A graph G is (k,0)‐colorable if its vertices can be partitioned into subsets V₁ and V₂ such that in G[V₁] every vertex has degree at most k, while G[V₂] is edgeless. For every integer k?0, we prove that every graph with the maximum average degree smaller than (3k+4)/(k+2) is (k,0)‐colorable. In particular, it follows that every planar graph with girth at least 7 is (8, 0)‐colorable. On the other hand, we construct planar graphs with girth 6 that are not (k,0)‐colorable for arbitrarily large k. © 2009 Wiley Periodicals, Inc. J Graph Theory 65:83–93, 2010     

Decreasing the diameter of bounded degree graphs

Let f_d (G) denote the minimum number of edges that have to be added to a graph G to transform it into a graph of diameter at most d. We prove that for any graph G with maximum degree D and n > n₀ (D) vertices, f₂(G) = n − D − 1 and f₃(G) ≥ n − O(D³). For d ≥ 4, f_d (G) depends strongly on the actual structure of G, not only on the maximum degree of G. We prove that the maximum of f_d (G) over all connected graphs on n vertices is n/⌊d/2 ⌋ − O(1). As a byproduct, we show that for the n‐cycle C_n, f_d (C_n) = n/(2⌊d/2 ⌋ − 1) − O(1) for every d and n, improving earlier estimates of Chung and Garey in certain ranges. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 161–172, 2000     

On the excess of vertex-transitive graphs of given degree and girth

We consider a restriction of the well-known Cage Problem to the class of vertex-transitive graphs, and consider the problem of finding the smallest vertex-transitive $k$-regular graphs of girth $g$. Counting cycles to obtain necessary arithmetic conditions on the parameters $(k,g)$, we extend previous results of Biggs, and prove that, for any given excess $e$ and any given degree $k\ge 4$, the asymptotic density of the set of girths $g$ for which there exists a vertex-transitive $(k,g)$-cage with excess not exceeding $e$ is 0.     

Total choosability of planar graphs with maximum degree 4

Let G be a planar graph with maximum degree 4. It is known that G is 8-totally choosable. It has been recently proved that if G has girth g?6, then G is 5-totally choosable. In this note we improve the first result by showing that G is 7-totally choosable and complete the latter one by showing that G is 6-totally choosable if G has girth at least 5.     

On the size of edge-coloring critical graphs with maximum degree 4

In 1968, Vizing proposed the following conjecture: If G=(V,E) is a Δ-critical graph of order n and size m, then . This conjecture has been verified for the cases of Δ≤5. In this paper, we prove that when Δ=4. It improves the known bound for Δ=4 when n>6.     

HSAGA and its application for the construction of near-Moore digraphs

The degree/diameter problem is to determine the largest graphs or digraphs of given maximum degree and given diameter. This paper deals with directed graphs. General upper bounds, called Moore bounds, exist for the largest possible order of such digraphs of maximum degree d and given diameter k. It is known that simulated annealing and genetic algorithm are effective techniques to identify global optimal solutions.This paper describes our attempt to build a Hybrid Simulated Annealing and Genetic Algorithm (HSAGA) that can be used to construct large digraphs. We present our new results obtained by HSAGA, as well as several related open problems.