On locally finite groups with a small centralizer of a four-subgroup

The main result of the paper is the following theorem. Let G be a locally finite group having a four-subgroup V such that C _G (V) is finite. Suppose that V contains two involutions v ₁ and v ₂ such that the centralizers C _G (v ₁) and C _G (v ₂) have finite exponent. Then G is almost locally soluble and [G, V]′ has finite exponent. Since [G, V] has finite index in G, the result gives a fairly detailed information about the structure of G.     

Exponent of a finite group with a four-group of automorphisms

Let A be a group isomorphic with either S ₄, the symmetric group on four symbols, or D ₈, the dihedral group of order 8. Let V be a normal four-subgroup of A and ?? an involution in ${A\setminus V}$ . Suppose that A acts on a finite group G in such a manner that C _G (V)?=?1 and C _G (??) has exponent e. We show that if ${A\cong S_4}$ then the exponent of G is e-bounded and if ${A\cong D_8}$ then the exponent of the derived group G?? is e-bounded. This work was motivated by recent results on the exponent of a finite group admitting an action by a Frobenius group of automorphisms.     

Edge degrees and dominating cycles

The edge degree d(e) of the edge e=uv is defined as the number of neighbours of e, i.e., |N(u)∪N(v)|-2. Two edges are called remote if they are disjoint and there is no edge joining them. In this article, we prove that in a 2-connected graph G, if d(e₁)+d(e₂)>|V(G)|-4 for any remote edges e₁,e₂, then all longest cycles C in G are dominating, i.e., G-V(C) is edgeless. This lower bound is best possible.As a corollary, it holds that if G is a 2-connected triangle-free graph with σ₂(G)>|V(G)|/2, then all longest cycles are dominating.     

New bounds for the broadcast domination number of a graph

A dominating broadcast on a graph G = (V, E) is a function f: V → {0, 1, ..., diam G} such that f(v) ≤ e(v) (the eccentricity of v) for all v ∈ V and such that each vertex is within distance f(v) from a vertex v with f(v) > 0. The cost of a broadcast f is σ(f) = Σ _v∈V f(v), and the broadcast number λ _b (G) is the minimum cost of a dominating broadcast. A set X ? V(G) is said to be irredundant if each x ∈ X dominates a vertex y that is not dominated by any other vertex in X; possibly y = x. The irredundance number ir (G) is the cardinality of a smallest maximal irredundant set of G. We prove the bound λ_b(G) ≤ 3 ir(G)/2 for any graph G and show that equality is possible for all even values of ir (G). We also consider broadcast domination as an integer programming problem, the dual of which provides a lower bound for λ_b.     

Balanced decomposition of a vertex-colored graph

A balanced vertex-coloring of a graph G is a function c from V(G) to {−1,0,1} such that ∑{c(v):v∈V(G)}=0. A subset U of V(G) is called a balanced set if U induces a connected subgraph and ∑{c(v):v∈U}=0. A decomposition V(G)=V₁∪?∪V_r is called a balanced decomposition if V_i is a balanced set for 1≤i≤r.In this paper, the balanced decomposition number f(G) of G is introduced; f(G) is the smallest integer s such that for any balanced vertex-coloring c of G, there exists a balanced decomposition V(G)=V₁∪?∪V_r with |V_i|≤s for 1≤i≤r. Balanced decomposition numbers of some basic families of graphs such as complete graphs, trees, complete bipartite graphs, cycles, 2-connected graphs are studied.     

Cycle-magic graphs

A simple graph G=(V,E) admits a cycle-covering if every edge in E belongs at least to one subgraph of G isomorphic to a given cycle C. Then the graph G is C-magic if there exists a total labelling f:V∪E→{1,2,…,|V|+|E|} such that, for every subgraph H^′=(V^′,E^′) of G isomorphic to C, ∑_v∈V^′f(v)+∑_e∈E^′f(e) is constant. When f(V)={1,…,|V|}, then G is said to be C-supermagic.We study the cyclic-magic and cyclic-supermagic behavior of several classes of connected graphs. We give several families of C_r-magic graphs for each r?3. The results rely on a technique of partitioning sets of integers with special properties.     

Orbits of the Actions of Odd-Order Solvable Groups

Suppose that V is a finite faithful irreducible G-module where G is a finite solvable group of odd order. We prove if the action is quasi-primitive, then either F(G) is abelian or G has at least 212 regular orbits on V. As an application, we prove that when V is a finite faithful completely reducible G-module for a solvable group G of odd order, then there exists v ∈ V such that C _G (v) ? F ₂(G) (where F ₂(G) is the 2nd ascending Fitting subgroup of G). We also generalize a result of Espuelas and Navarro. Let G be a group of odd order and let H be a Hall π-subgroup of G. Let V be a faithful G-module over a finite field of characteristic 2, then there exists v ∈ V such that C _H (v) ? O _π(G).     

Large orbits of elements centralized by a Sylow subgroup

If G has a nilpotent normal p-complement and V is a finite, faithful and completely reducible G-module of characteristic p, we prove that there exist ${v_1, v_2 \in V}If G has a nilpotent normal p-complement and V is a finite, faithful and completely reducible G-module of characteristic p, we prove that there exist v₁, v₂ ? V{v_1, v_2 \in V} such that C_G(v₁)?C_G(v₂) = P{{\bf C}_{G}{(v_1)}\cap {\bf C}_{G}{(v_2)} = P} , where P ? Syl_p(G){P \in {\rm Syl}_p(G)} . We hence deduce that, if the normal p-complement K is nontrivial, there exists v ? C_V(P){v \in {\bf C}_{V}(P)} such that |K : C _K (v)|² > |K|.     

The eccentric connectivity index of nanotubes and nanotori

Let G be a molecular graph. The eccentric connectivity index ξ^c(G) is defined as ξ^c(G)=∑_u∈V(G)deg_G(u)ε_G(u), where deg_G(u) denotes the degree of vertex u and ε_G(u) is the largest distance between u and any other vertex v of G. In this paper exact formulas for the eccentric connectivity index of TUC₄C₈(S) nanotube and TC₄C₈(S) nanotorus are given.     

Lower negative decision number in a graph

Let G=(V,E) be a graph. A function f:V(G)→{?1,1} is called bad if ∑ _v∈N(v) f(v)≤1 for every v∈V(G). A bad function f of a graph G is maximal if there exists no bad function g such that g≠f and g(v)≥f(v) for every v∈V. The minimum of the values of ∑ _v∈V f(v), taken over all maximal bad functions f, is called the lower negative decision number and is denoted by β _D ^* (G). In this paper, we present sharp lower bounds on this number for regular graphs and nearly regular graphs, and we also characterize the graphs attaining those bounds.     

On a cyclic connectivity property of directed graphs

Let us call a digraph D cycle-connected if for every pair of vertices u,v∈V(D) there exists a cycle containing both u and v. In this paper we study the following open problem introduced by Ádám. Let D be a cycle-connected digraph. Does there exist a universal edge in D, i.e., an edge e∈E(D) such that for every w∈V(D) there exists a cycle C such that w∈V(C) and e∈E(C)?In his 2001 paper Hetyei conjectured that cycle-connectivity always implies the existence of a universal edge. In the present paper we prove the conjecture of Hetyei for bitournaments.     

Signed star domatic number of a graph

Let G be a simple graph without isolated vertices with vertex set V(G) and edge set E(G). A function f:E(G)?{−1,1} is said to be a signed star dominating function on G if ∑_e∈E(v)f(e)≥1 for every vertex v of G, where E(v)={uv∈E(G)∣u∈N(v)}. A set {f₁,f₂,…,f_d} of signed star dominating functions on G with the property that for each e∈E(G), is called a signed star dominating family (of functions) on G. The maximum number of functions in a signed star dominating family on G is the signed star domatic number of G, denoted by d_SS(G).In this paper we study the properties of the signed star domatic number d_SS(G). In particular, we determine the signed domatic number of some classes of graphs.     

Vertex-disjoint quadrilaterals containing specified edges in a bipartite graph

The theory of vertex-disjoint cycles and 2-factors of graphs is the extension and generation of the well-known Hamiltonian cycles theory and it has important applications in network communication. In this paper we first prove the following result. Let G=(V ₁,V ₂;E) be a bipartite graph with |V ₁|=|V ₂|=n such that n≥2k+1, where k≥1 is an integer. If d(x)+d(y)≥?(4n+2k?1)/3? for each pair of nonadjacent vertices x and y of G with x∈V ₁ and y∈V ₂, then, for any k independent edges e ₁,…,e _k of G, G contains k vertex-disjoint quadrilaterals C ₁,…,C _k such that e _i ∈E(C _i ) for each i∈{1,…,k}. We further show that the degree condition above is sharp. If |V ₁|=|V ₂|=2k, we give degree conditions that G has a 2-factor with k vertex-disjoint quadrilaterals C ₁,…,C _k containing specified edges of G.     

A note on traversing specified vertices in graphs embedded with large representativity

A graph G is said to have property P(2,k) if given any k+2 distinct vertices a,b,v₁,…,v_k, there is a path P in G joining a and b and passing through all of v₁,…,v_k. A graph G is said to have property C(k) if given any k distinct vertices v₁,…,v_k, there is a cycle C in G containing all of v₁,…,v_k. It is shown that if a 4-connected graph G is embedded in an orientable surface Σ (other than the sphere) of Euler genus eg(G,Σ), with sufficiently large representativity (as a function of both eg(G,Σ) and k), then G possesses both properties P(2,k) and C(k).     

A note on balanced bipartitions

A balanced bipartition of a graph G is a bipartition V₁ and V₂ of V(G) such that −1≤|V₁|−|V₂|≤1. Bollobás and Scott conjectured that if G is a graph with m edges and minimum degree at least 2 then G admits a balanced bipartition V₁,V₂ such that max{e(V₁),e(V₂)}≤m/3, where e(V_i) denotes the number of edges of G with both ends in V_i. In this note, we prove this conjecture for graphs with average degree at least 6 or with minimum degree at least 5. Moreover, we show that if G is a graph with m edges and n vertices, and if the maximum degree Δ(G)=o(n) or the minimum degree δ(G)→∞, then G admits a balanced bipartition V₁,V₂ such that max{e(V₁),e(V₂)}≤(1+o(1))m/4, answering a question of Bollobás and Scott in the affirmative. We also provide a sharp lower bound on max{e(V₁,V₂):V₁,V₂ is a balanced bipartition of G}, in terms of size of a maximum matching, where e(V₁,V₂) denotes the number of edges between V₁ and V₂.     

Small cycles and 2-factor passing through any given vertices in graphs

The theory of vertex-disjoint cycles and 2-factor of graphs has important applications in computer science and network communication. For a graph G, let σ ₂(G):=min?{d(u)+d(v)|uv ? E(G),u≠v}. In the paper, the main results of this paper are as follows:

1. Let k≥2 be an integer and G be a graph of order n≥3k, if σ ₂(G)≥n+2k?2, then for any set of k distinct vertices v ₁,…,v _k , G has k vertex-disjoint cycles C ₁,C ₂,…,C _k of length at most four such that v _i ∈V(C _i ) for all 1≤i≤k.
2. Let k≥1 be an integer and G be a graph of order n≥3k, if σ ₂(G)≥n+2k?2, then for any set of k distinct vertices v ₁,…,v _k , G has k vertex-disjoint cycles C ₁,C ₂,…,C _k such that:
   1. v _i ∈V(C _i ) for all 1≤i≤k.
   2. V(C ₁)∪???∪V(C _k )=V(G), and
   3. |C _i |≤4, 1≤i≤k?1.

Moreover, the condition on σ ₂(G)≥n+2k?2 is sharp.     

Extremal perfect graphs for a bound on the domination number

We examine classes of extremal graphs for the inequality γ(G)?|V|-max{d(v)+β_v(G)}, where γ(G) is the domination number of graph G, d(v) is the degree of vertex v, and β_v(G) is the size of a largest matching in the subgraph of G induced by the non-neighbours of v. This inequality improves on the classical upper bound |V|-maxd(v) due to Claude Berge. We give a characterization of the bipartite graphs and of the chordal graphs that achieve equality in the inequality. The characterization implies that the extremal bipartite graphs can be recognized in polynomial time, while the corresponding problem remains NP-complete for the extremal chordal graphs.     

Every graph is (2,3)-choosable

A total weighting of a graph G is a mapping ? that assigns to each element z ∈ V (G)∪E(G) a weight ?(z). A total weighting ? is proper if for any two adjacent vertices u and v, ∑ _e∈E(u) ?(e)+?(u)≠∑ _e∈E(v) ?(e)+?(v). This paper proves that if each edge e is given a set L(e) of 3 permissible weights, and each vertex v is given a set L(v) of 2 permissible weights, then G has a proper total weighting ? with ?(z) ∈ L(z) for each element z ∈ V (G)∪E(G).     

On periodic solvable groups having automorphisms with nilpotent fixed point groups

Letp be a prime,G a periodic solvablep′-group acted on by an elementary groupV of orderp ². We show that ifC _G(v) is abelian for eachv ∈V ^# thenG has nilpotent derived group, and ifp=2 andC _G(v) is nilpotent for eachv ∈V ^# thenG is metanilpotent. Earlier results of this kind were known only for finite groups.