The Independence Number for De Bruijn networks and Kautz networks

Let D be a directed graph; the (l,ω)-Independence Number of graph D, denoted by α_l,ω(D), is an important performance parameter for interconnection networks. De Bruijn networks and Kautz networks, denoted by B(d,n) and K(d,n) respectively, are versatile and efficient topological structures of interconnection networks. For l=1,2,…,n, this paper shows that α_l,d−1(B(d,n))=dⁿ,α_l,d−1(K(d,n))=α_l,d(K(d,n))=dⁿ+dⁿ⁻¹ if d≥3 and n≤d−2. In particular, the paper shows the exact value of the Independence Number for B(d,1) and B(d,2) for any d. For the generalized situation, the paper obtains a lower bound α_l,d−1(B(d,n))≥d² if n≥3 and d≥5.     

Improvement on Brooks' chromatic bound for a class of graphs

Brooks' Theorem says that if for a graph G,Δ(G)=n, then G is n-colourable, unless (1) n=2 and G has an odd cycle as a component, or (2) n>2 and K_n+1 is a component of G. In this paper we prove that if a graph G has none of some three graphs (K_1,3;K₅?e and H) as an induced subgraph and if Δ(G)?6 and d(G)<Δ(G), then χ(G)<Δ(G). Also we give examples to show that the hypothesis Δ(G)?6 can not be non-trivially relaxed and the graph K₅?e can not be removed from the hypothesis. Moreover, for a graph G with none of K_1,3;K₅?e and H as an induced subgraph, we verify Borodin and Kostochka's conjecture that if for a graph G,Δ(G)?9 and d(G)<Δ(G), then χ(G)<Δ(G).     

Uniform boundary Harnack principle and generalized triangle property

Let D be a bounded open subset in R^d, d?2, and let G denote the Green function for D with respect to (-Δ)^α/2, 0<α?2, α<d. If α<2, assume that D satisfies the interior corkscrew condition; if α=2, i.e., if G is the classical Green function on D, assume—more restrictively—that D is a uniform domain. Let g=G(·,y₀)∧1 for some y₀∈D. Based on the uniform boundary Harnack principle, it is shown that G has the generalized triangle property which states that when d(z,x)?d(z,y). An intermediate step is the approximation G(x,y)≈|x-y|^α-dg(x)g(y)/g(A)², where A is an arbitrary point in a certain set B(x,y).This is discussed in a general setting where D is a dense open subset of a compact metric space satisfying the interior corkscrew condition and G is a quasi-symmetric positive numerical function on D×D which has locally polynomial decay and satisfies Harnack's inequality. Under these assumptions, the uniform boundary Harnack principle, the approximation for G, and the generalized triangle property turn out to be equivalent.     

The edge-face coloring of graphs embedded in a surface of characteristic zero

Let G be a graph embedded in a surface of characteristic zero with maximum degree Δ. The edge-face chromatic number χ_ef(G) of G is the least number of colors such that any two adjacent edges, adjacent faces, incident edge and face have different colors. In this paper, we prove that χ_ef(G)≤Δ+1 if Δ≥13, χ_ef(G)≤Δ+2 if Δ≥12, χ_ef(G)≤Δ+3 if Δ≥4, and χ_ef(G)≤7 if Δ≤3.     

Density of sets of natural numbers and the Lévy group

Let N denote the set of positive integers. The asymptotic density of the set A⊆N is d(A)=limn→∞|A∩[1,n]|/n, if this limit exists. Let AD denote the set of all sets of positive integers that have asymptotic density, and let SN denote the set of all permutations of the positive integers N. The group L^? consists of all permutations f∈SN such that A∈AD if and only if f(A)∈AD, and the group L^* consists of all permutations f∈L^? such that d(f(A))=d(A) for all A∈AD. Let be a one-to-one function such that d(f(N))=1 and, if A∈AD, then f(A)∈AD. It is proved that f must also preserve density, that is, d(f(A))=d(A) for all A∈AD. Thus, the groups L^? and L^* coincide.     

Planar graphs without 5-cycles or without 6-cycles

Let G be a planar graph without 5-cycles or without 6-cycles. In this paper, we prove that if G is connected and δ(G)≥2, then there exists an edge xy∈E(G) such that d(x)+d(y)≤9, or there is a 2-alternating cycle. By using the above result, we obtain that (1) its linear 2-arboricity , (2) its list total chromatic number is Δ(G)+1 if Δ(G)≥8, and (3) its list edge chromatic number is Δ(G) if Δ(G)≥8.     

On distance two graphs of upper bound graphs

For a poset P=(X,≤), the upper bound graph (UB-graph) of P is the graph U=(X,E_U), where uv∈E_U if and only if u≠v and there exists m∈X such that u,v≤m. For a graph G, the distance two graph DS₂(G) is the graph with vertex set V(DS₂(G))=V(G) and u,v∈V(DS₂(G)) are adjacent if and only if d_G(u,v)=2. In this paper, we deal with distance two graphs of upper bound graphs. We obtain a characterization of distance two graphs of split upper bound graphs.     

Grundy number and products of graphs

The Grundy number of a graph G, denoted by Γ(G), is the largest k such that G has a greedyk-colouring, that is a colouring with k colours obtained by applying the greedy algorithm according to some ordering of the vertices of G. In this paper, we study the Grundy number of the lexicographic and cartesian products of two graphs in terms of the Grundy numbers of these graphs.Regarding the lexicographic product, we show that Γ(G)×Γ(H)≤Γ(G[H])≤2^Γ(G)−1(Γ(H)−1)+Γ(G). In addition, we show that if G is a tree or Γ(G)=Δ(G)+1, then Γ(G[H])=Γ(G)×Γ(H). We then deduce that for every fixed c≥1, given a graph G, it is CoNP-Complete to decide if Γ(G)≤c×χ(G) and it is CoNP-Complete to decide if Γ(G)≤c×ω(G).Regarding the cartesian product, we show that there is no upper bound of Γ(G□H) as a function of Γ(G) and Γ(H). Nevertheless, we prove that Γ(G□H)≤Δ(G)⋅2^Γ(H)−1+Γ(H).     

Chromatic coloring with a maximum color class

Let G be any graph, and also let Δ(G), χ(G) and α(G) denote the maximum degree, the chromatic number and the independence number of G, respectively. A chromatic coloring of G is a proper coloring of G using χ(G) colors. A color class in a proper coloring of G is maximum if it has size α(G). In this paper, we prove that if a graph G (not necessarily connected) satisfies χ(G)≥Δ(G), then there exists a chromatic coloring of G in which some color class is maximum. This cannot be guaranteed if χ(G)<Δ(G). We shall also give some other extensions.     

On algebraic connectivity augmentation

Suppose that G is an undirected graph, and that H is a spanning subgraph of G^c whose edges induce a subgraph on p vertices. We consider the expression α(G∪H)-α(G), where α denotes the algebraic connectivity. Specifically, we provide upper and lower bounds on α(G∪H)-α(G) in terms of p, and characterise the corresponding equality cases. We also discuss the density of the expression α(G∪H)-α(G) in the interval [0,p]. A bound on α(G∪H)-α(G) is provided in a special case, and several examples are considered.     

Hamiltonicity and pancyclicity of cartesian products of graphs

The cartesian product of a graph G with K₂ is called a prism over G. We extend known conditions for hamiltonicity and pancyclicity of the prism over a graph G to the cartesian product of G with paths, cycles, cliques and general graphs. In particular we give results involving cubic graphs and almost claw-free graphs.We also prove the following: Let G and H be two connected graphs. Let both G and H have a 2-factor. If Δ(G)≤g^′(H) and Δ(H)≤g^′(G) (we denote by g^′(F) the length of a shortest cycle in a 2-factor of a graph F taken over all 2-factorization of F), then G□H is hamiltonian.     

Connectivity of the Mycielskian of a graph

In a search for triangle-free graphs with arbitrarily large chromatic numbers, Mycielski developed a graph transformation that transforms a graph G into a new graph μ(G), which is called the Mycielskian of G. This paper investigates the vertex-connectivity κ(μ(G)) and edge-connectivity κ^′(μ(G)) of μ(G) . We show that κ(μ(G))=min{δ(μ(G)),2κ(G)+1} and κ^′(μ(G))=δ(μ(G)).     

On Optimal Orientations of Cartesian Products of Trees

. Let d(D) (resp., d(G)) denote the diameter and r(D) (resp., r(G)) the radius of a digraph D (resp., graph G). Let G×H denote the cartesian product of two graphs G and H. An orientation D of G is said to be (r, d)-invariant if r(D)=r(G) and d(D)=d(G). Let {T _i }, i=1,…,n, where n≥2, be a family of trees. In this paper, we show that the graph ∏ _{i =1} ⁿ T _i admits an (r, d)-invariant orientation provided that d(T ₁)≥d(T ₂)≥4 for n=2, and d(T ₁)≥5 and d(T ₂)≥4 for n≥3. Received: July 30, 1997 Final version received: April 20, 1998     

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 an interpolation property of outerplanar graphs

Let D be an acyclic orientation of a graph G. An arc of D is said to be dependent if its reversal creates a directed cycle. Let d(D) denote the number of dependent arcs in D. Define d_min(G) (d_max(G)) to be the minimum (maximum) number of d(D) over all acyclic orientations D of G. We determine d_min(G) for an outerplanar graph G and prove that G has an acyclic orientation with exactly k dependent arcs if d_min(G)?k?d_max(G).     

Eigenvalues of a graph and its imbeddings

A graph H is imbedded in a graph G if a subset of the vertices of G determines a subgraph isomorphic to H. If λ(G) is the least eigenvalue of G and k_R(H) = lim sup_d→∞ {λ(G)| H imbedded in G; G regular and connected; diam(G) > d; deg(G) > d}, then λ(H) ? 2 ≤ k_R(H) ≤ λ(H) with these bounds being the best possible. Given a graph H, there exist arbitrarily large families of isospectral graphs such that H can be imbedded in each member of the family.     

Independence and coloring properties of direct products of some vertex-transitive graphs

Let α(G) and χ(G) denote the independence number and chromatic number of a graph G, respectively. Let G×H be the direct product graph of graphs G and H. We show that if G and H are circular graphs, Kneser graphs, or powers of cycles, then α(G×H)=max{α(G)|V(H)|,α(H)|V(G)|} and χ(G×H)=min{χ(G),χ(H)}.     

Topology of random clique complexes

In a seminal paper, Erd?s and Rényi identified a sharp threshold for connectivity of the random graph G(n,p). In particular, they showed that if p?logn/n then G(n,p) is almost always connected, and if p?logn/n then G(n,p) is almost always disconnected, as n→∞.The clique complexX(H) of a graph H is the simplicial complex with all complete subgraphs of H as its faces. In contrast to the zeroth homology group of X(H), which measures the number of connected components of H, the higher dimensional homology groups of X(H) do not correspond to monotone graph properties. There are nevertheless higher dimensional analogues of the Erd?s-Rényi Theorem.We study here the higher homology groups of X(G(n,p)). For k>0 we show the following. If p=n^α, with α<−1/k or α>−1/(2k+1), then the kth homology group of X(G(n,p)) is almost always vanishing, and if −1/k<α<−1/(k+1), then it is almost always nonvanishing.We also give estimates for the expected rank of homology, and exhibit explicit nontrivial classes in the nonvanishing regime. These estimates suggest that almost all d-dimensional clique complexes have only one nonvanishing dimension of homology, and we cannot rule out the possibility that they are homotopy equivalent to wedges of a spheres.     

A short and unified proof of Yu et al.?s two results on the eccentric distance sum

The eccentric distance sum (EDS) is a novel topological index that offers a vast potential for structure activity/property relationships. For a connected graph G, the eccentric distance sum is defined as ξd(G)=_∑v∈V(G)ecG(v)DG(v), where ecG(v) is the eccentricity of a vertex v in G and DG(v) is the sum of distances of all vertices in G from v. More recently, Yu et al. [G. Yu, L. Feng, A. Ili?, On the eccentric distance sum of trees and unicyclic graphs, J. Math. Anal. Appl. 375 (2011) 99-107] proved that for an n-vertex tree T, ξd(T)?4n²−9n+5, with equality holding if and only if T is the n-vertex star Sn, and for an n-vertex unicyclic graph G, ξd(G)?4n²−9n+1, with equality holding if and only if G is the graph obtained by adding an edge between two pendent vertices of n-vertex star. In this note, we give a short and unified proof of the above two results.     

Completing partial packings of bipartite graphs

Given a bipartite graph H and an integer n, let f(n;H) be the smallest integer such that any set of edge disjoint copies of H on n vertices can be extended to an H-design on at most n+f(n;H) vertices. We establish tight bounds for the growth of f(n;H) as n→∞. In particular, we prove the conjecture of Füredi and Lehel (2010) [4] that f(n;H)=o(n). This settles a long-standing open problem.