Bipartite graphs are not universal fixers

For any permutation π of the vertex set of a graph G, the graph πG is obtained from two copies G^′ and G^″ of G by joining u∈V(G^′) and v∈V(G^″) if and only if v=π(u). Denote the domination number of G by γ(G). For all permutations π of V(G), γ(G)≤γ(πG)≤2γ(G). If γ(πG)=γ(G) for all π, then G is called a universal fixer. We prove that graphs without 5-cycles are not universal fixers, from which it follows that bipartite graphs are not universal fixers.     

Regular graphs are not universal fixers

For any permutation π of the vertex set of a graph G, the graph πG is obtained from two copies G^′ and G^″ of G by joining u∈V(G^′) and v∈V(G^″) if and only if v=π(u). Denote the domination number of G by γ(G). For all permutations π of V(G), γ(G)≤γ(πG)≤2γ(G). If γ(πG)=γ(G) for all π, then G is called a universal fixer. We prove that regular graphs and graphs with γ=4 are not universal fixers.     

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.     

Domination critical graphs

A set of vertices S is said to dominate the graph G if for each v ? S, there is a vertex u ∈ S with u adjacent to v. The smallest cardinality of any such dominating set is called the domination number of G and is denoted by γ(G). The purpose of this paper is to initiate an investigation of those graphs which are critical in the following sense: For each v, u ∈ V(G) with v not adjacent to u, γ(G + vu) < γ(G). Thus G is k-y-critical if γ(G) = k and for each edge e ? E(G), γ(G + e) = k ?1. The 2-domination critical graphs are characterized the properties of the k-critical graphs with k ≥ 3 are studied. In particular, the connected 3-critical graphs of even order are shown to have a 1-factor and some stringent restrictions on their degree sequences and diameters are obtained.     

On a conjecture of the Randi? index

The Randi? index of a graph G is defined as , where d(u) is the degree of vertex u and the summation goes over all pairs of adjacent vertices u, v. A conjecture on R(G) for connected graph G is as follows: R(G)≥r(G)−1, where r(G) denotes the radius of G. We proved that the conjecture is true for biregular graphs, connected graphs with order n≤10 and tricyclic graphs.     

Hamiltonian connectedness in 3-connected line graphs

We investigate graphs G such that the line graph L(G) is hamiltonian connected if and only if L(G) is 3-connected, and prove that if each 3-edge-cut contains an edge lying in a short cycle of G, then L(G) has the above mentioned property. Our result extends Kriesell’s recent result in [M. Kriesell, All 4-connected line graphs of claw free graphs are hamiltonian-connected, J. Combin. Theory Ser. B 82 (2001) 306-315] that every 4-connected line graph of a claw free graph is hamiltonian connected. Another application of our main result shows that if L(G) does not have an hourglass (a graph isomorphic to K₅−E(C₄), where C₄ is an cycle of length 4 in K₅) as an induced subgraph, and if every 3-cut of L(G) is not independent, then L(G) is hamiltonian connected if and only if κ(L(G))≥3, which extends a recent result by Kriesell [M. Kriesell, All 4-connected line graphs of claw free graphs are hamiltonian-connected, J. Combin. Theory Ser. B 82 (2001) 306-315] that every 4-connected hourglass free line graph is hamiltonian connected.     

Balanced and 1-balanced graph constructions

There are several density functions for graphs which have found use in various applications. In this paper, we examine two of them, the first being given by b(G)=|E(G)|/|V(G)|, and the other being given by g(G)=|E(G)|/(|V(G)|−ω(G)), where ω(G) denotes the number of components of G. Graphs for which b(H)≤b(G) for all subgraphs H of G are called balanced graphs, and graphs for which g(H)≤g(G) for all subgraphs H of G are called 1-balanced graphs (also sometimes called strongly balanced or uniformly dense in the literature). Although the functions b and g are very similar, they distinguish classes of graphs sufficiently differently that b(G) is useful in studying random graphs, g(G) has been useful in designing networks with reduced vulnerability to attack and in studying the World Wide Web, and a similar function is useful in the study of rigidity. First we give a new characterization of balanced graphs. Then we introduce a graph construction which generalizes the Cartesian product of graphs to produce what we call a generalized Cartesian product. We show that generalized Cartesian product derived from a tree and 1-balanced graphs are 1-balanced, and we use this to prove that the generalized Cartesian products derived from 1-balanced graphs are 1-balanced.     

Characterizing subgraphs of Hamming graphs

Cartesian products of complete graphs are known as Hamming graphs. Using embeddings into Cartesian products of quotient graphs we characterize subgraphs, induced subgraphs, and isometric subgraphs of Hamming graphs. For instance, a graph G is an induced subgraph of a Hamming graph if and only if there exists a labeling of E(G) fulfilling the following two conditions: (i) edges of a triangle receive the same label; (ii) for any vertices u and v at distance at least two, there exist two labels which both appear on any induced u, υ‐path. © 2005 Wiley Periodicals, Inc. J Graph Theory 49: 302–312, 2005     

Total edge irregularity strength of complete graphs and complete bipartite graphs

A total edge irregular k-labelling ν of a graph G is a labelling of the vertices and edges of G with labels from the set {1,…,k} in such a way that for any two different edges e and f their weights φ(f) and φ(e) are distinct. Here, the weight of an edge g=uv is φ(g)=ν(g)+ν(u)+ν(v), i. e. the sum of the label of g and the labels of vertices u and v. The minimum k for which the graph G has an edge irregular total k-labelling is called the total edge irregularity strength of G.We have determined the exact value of the total edge irregularity strength of complete graphs and complete bipartite graphs.     

Comparing the Zagreb indices for graphs with small difference between the maximum and minimum degrees

The first Zagreb index M₁(G) and the second Zagreb index M₂(G) of a (molecular) graph G are defined as M₁(G)=∑_u∈V(G)(d(u))² and M₂(G)=∑_uv∈E(G)d(u)d(v), where d(u) denotes the degree of a vertex u in G. The AutoGraphiX system [M. Aouchiche, J.M. Bonnefoy, A. Fidahoussen, G. Caporossi, P. Hansen, L. Hiesse, J. Lacheré, A. Monhait, Variable neighborhood search for extremal graphs. 14. The AutoGraphiX 2 system, in: L. Liberti, N. Maculan (Eds.), Global Optimization: From Theory to Implementation, Springer, 2005; G. Caporossi, P. Hansen, Variable neighborhood search for extremal graphs: 1 The AutoGraphiX system, Discrete Math. 212 (2000) 29-44; G. Caporossi, P. Hansen, Variable neighborhood search for extremal graphs. 5. Three ways to automate finding conjectures, Discrete Math. 276 (2004) 81-94] conjectured that M₁/n≤M₂/m (where n=|V(G)| and m=|E(G)|) for simple connected graphs. Hansen and Vuki?evi? [P. Hansen, D. Vuki?evi?, Comparing the Zagreb indices, Croat. Chem. Acta 80 (2007) 165-168] proved that it is true for chemical graphs and it does not hold for all graphs. Vuki?evi? and Graovac [D. Vuki?evi?, A. Graovac, Comparing Zagreb M₁ and M₂ indices for acyclic molecules, MATCH Commun. Math. Comput. Chem. 57 (2007) 587-590] proved that it is also true for trees. In this paper, we show that M₁/n≤M₂/m holds for graphs with Δ(G)−δ(G)≤2 and characterize the extremal graphs, the proof of which implies the result in [P. Hansen, D. Vuki?evi?, Comparing the Zagreb indices, Croat. Chem. Acta 80 (2007) 165-168]. We also obtain the result that M₁/n≤M₂/m holds for graphs with Δ(G)−δ(G)≤3 and δ(G)≠2.     

On Properties of a Graph that Depend on its Distance Function

If G is a connected graph with distance function d, then by a step in G is meant an ordered triple (u, x, v) of vertices of G such that d(u, x) = 1 and d(u, v) = d(x, v) + 1. A characterization of the set of all steps in a connected graph was published by the present author in 1997. In Section 1 of this paper, a new and shorter proof of that characterization is presented. A stronger result for a certain type of connected graphs is proved in Section 2.     

Diameter and connectivity of 3-arc graphs

An arc of a graph is an oriented edge and a 3-arc is a 4-tuple (v,u,x,y) of vertices such that both (v,u,x) and (u,x,y) are paths of length two. The 3-arc graph of a given graph G, X(G), is defined to have vertices the arcs of G. Two arcs uv,xy are adjacent in X(G) if and only if (v,u,x,y) is a 3-arc of G. This notion was introduced in recent studies of arc-transitive graphs. In this paper we study diameter and connectivity of 3-arc graphs. In particular, we obtain sharp bounds for the diameter and connectivity of X(G) in terms of the corresponding invariant of G.     

Domination of generalized Cartesian products

The generalized prism πG of G is the graph consisting of two copies of G, with edges between the copies determined by a permutation π acting on the vertices of G. We define a generalized Cartesian product that corresponds to the Cartesian product when π is the identity, and the generalized prism when H is the graph K₂. Burger, Mynhardt and Weakley [A.P. Burger, C.M. Mynhardt, W.D. Weakley, On the domination number of prisms of graphs, Discuss. Math. Graph Theory 24 (2) (2004) 303-318.] characterized universal doublers, i.e. graphs for which γ(πG)=2γ(G) for any π. In general for any n≥2 and permutation π, and a graph attaining equality in this upper bound for all π is called a universal multiplier. We characterize such graphs.     

Acyclic 5-choosability of planar graphs without 4-cycles

A proper vertex coloring of a graph G=(V,E) is acyclic if G contains no bicolored cycle. A graph G is acyclically L-list colorable if for a given list assignment L={L(v):v∈V}, there exists a proper acyclic coloring π of G such that π(v)∈L(v) for all v∈V. If G is acyclically L-list colorable for any list assignment with |L(v)|≥k for all v∈V, then G is acyclically k-choosable. In this paper we prove that every planar graph without 4-cycles and without two 3-cycles at distance less than 3 is acyclically 5-choosable. This improves a result in [M. Montassier, P. Ochem, A. Raspaud, On the acyclic choosability of graphs, J. Graph Theory 51 (2006) 281-300], which says that planar graphs of girth at least 5 are acyclically 5-choosable.     

Secure domination critical graphs

A secure dominating set X of a graph G is a dominating set with the property that each vertex u∈V_G−X is adjacent to a vertex v∈X such that (X−{v})∪{u} is dominating. The minimum cardinality of such a set is called the secure domination number, denoted by γ_s(G). We are interested in the effect of edge removal on γ_s(G), and characterize γ_s-ER-critical graphs, i.e. graphs for which γ_s(G−e)>γ_s(G) for any edge e of G, bipartite γ_s-ER-critical graphs and γ_s-ER-critical trees.     

Lower bounds for the domination number and the total domination number of direct product graphs

A sharp lower bound for the domination number and the total domination number of the direct product of finitely many complete graphs is given: . Sharpness is established in the case when the factors are large enough in comparison to the number of factors. The main result gives a lower bound for the domination (and the total domination) number of the direct product of two arbitrary graphs: γ(G×H)≥γ(G)+γ(H)−1. Infinite families of graphs that attain the bound are presented. For these graphs it also holds that γ_t(G×H)=γ(G)+γ(H)−1. Some additional parallels with the total domination number are made.     

Equivariant collapses and the homotopy type of iterated clique graphs

The clique graph K(G) of a simple graph G is the intersection graph of its maximal complete subgraphs, and we define iterated clique graphs by K⁰(G)=G, Kⁿ⁺¹(G)=K(Kⁿ(G)). We say that two graphs are homotopy equivalent if their simplicial complexes of complete subgraphs are so. From known results, it can be easily inferred that Kⁿ(G) is homotopy equivalent to G for every n if G belongs to the class of clique-Helly graphs or to the class of dismantlable graphs. However, in both of these cases the collection of iterated clique graphs is finite up to isomorphism. In this paper, we show two infinite classes of clique-divergent graphs that satisfy G?Kⁿ(G) for all n, moreover Kⁿ(G) and G are simple-homotopy equivalent. We provide some results on simple-homotopy type that are of independent interest.     

A constructive characterization of total domination vertex critical graphs

Let G be a graph of order n and maximum degree Δ(G) and let γ_t(G) denote the minimum cardinality of a total dominating set of a graph G. A graph G with no isolated vertex is the total domination vertex critical if for any vertex v of G that is not adjacent to a vertex of degree one, the total domination number of G−v is less than the total domination number of G. We call these graphs γ_t-critical. For any γ_t-critical graph G, it can be shown that n≤Δ(G)(γ_t(G)−1)+1. In this paper, we prove that: Let G be a connected γ_t-critical graph of order n (n≥3), then n=Δ(G)(γ_t(G)−1)+1 if and only if G is regular and, for each v∈V(G), there is an A⊆V(G)−{v} such that N(v)∩A=0?, the subgraph induced by A is 1-regular, and every vertex in V(G)−A−{v} has exactly one neighbor in A.     

Local coloring of Kneser graphs

A local coloring of a graph G is a function c:V(G)→N having the property that for each set S⊆V(G) with 2≤|S|≤3, there exist vertices u,v∈S such that |c(u)−c(v)|≥m_S, where m_S is the number of edges of the induced subgraph 〈S〉. The maximum color assigned by a local coloring c to a vertex of G is called the value of c and is denoted by χ_?(c). The local chromatic number of G is χ_?(G)=min{χ_?(c)}, where the minimum is taken over all local colorings c of G. The local coloring of graphs was introduced by Chartrand et al. [G. Chartrand, E. Salehi, P. Zhang, On local colorings of graphs, Congressus Numerantium 163 (2003) 207-221]. In this paper the local coloring of Kneser graphs is studied and the local chromatic number of the Kneser graph K(n,k) for some values of n and k is determined.     

Independent sets in tensor graph powers

The tensor product of two graphs, G and H, has a vertex set V(G) × V(H) and an edge between (u,v) and (u′,v′) iff both u u′ ∈ E(G) and v v′ ∈ E(H). Let A(G) denote the limit of the independence ratios of tensor powers of G, lim, α(Gⁿ)/|V(Gⁿ)|. This parameter was introduced in [Brown, Nowakowski, Rall, SIAM J Discrete Math 9 ( 5 ), 290–300], where it was shown that A(G) is lower bounded by the vertex expansion ratio of independent sets of G. In this article we study the relation between these parameters further, and ask whether they are in fact equal. We present several families of graphs where equality holds, and discuss the effect the above question has on various open problems related to tensor graph products. © 2006 Wiley Periodicals, Inc. J Graph Theory