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.     

On Zero-Sum {\mathbb{Z}_k} -Magic Labelings of 3-Regular Graphs

Let G =  (V, E) be a finite loopless graph and let (A, +) be an abelian group with identity 0. Then an A-magic labeling of G is a function ${\phi}$ from E into A ? {0} such that for some ${a \in A, \sum_{e \in E(v)} \phi(e) = a}$ for every ${v \in V}$ , where E(v) is the set of edges incident to v. If ${\phi}$ exists such that a =  0, then G is zero-sum A-magic. Let zim(G) denote the subset of ${\mathbb{N}}$ (the positive integers) such that ${1 \in zim(G)}$ if and only if G is zero-sum ${\mathbb{Z}}$ -magic and ${k \geq 2 \in zim(G)}$ if and only if G is zero-sum ${\mathbb{Z}_k}$ -magic. We establish that if G is 3-regular, then ${zim(G) = \mathbb{N} - \{2\}}$ or ${\mathbb{N} - \{2,4\}.}$      

2-Reducible Two Paths and an Edge Constructing a Path in (2k + 1)-Edge-Connected Graphs

Let k ≥ 5 be an odd integer and G = (V(G), E(G)) be a k-edge-connected graph. For ${X\subseteq V(G),e(X)}$ denotes the number of edges between X and V(G) ? X. We here prove that if ${\{s_i,t_i\}\subseteq X_i\subseteq V(G)(i=1,2),f}$ is an edge between s ₁ and ${s_2,X_1\cap X_2=\emptyset,e(X_1)\le 2k-3,e(X_2)\le 2k-2}$ , and e(Y) ≥ k + 1 for each ${Y\subseteq V(G)}$ with ${Y\cap\{s_1,t_1,s_2,t_2\}=\{s_1,t_2\}}$ , then there exist paths P ₁ and P ₂ such that P _i joins s _i and ${t_i,V(P_i)\subseteq X_i}$ (i = 1, 2) and ${G-f-E(P_1\cup P_2)}$ is (k ? 2)-edge-connected, and in fact we give a generalization of this result.     

On conjugate characteristic polynomial of a graph

Let $c=a+b\sqrt{m}$ and $\overline{c}=a-b\sqrt{m}$ , where a and b are two nonzero integers and m is a positive integer such that m is not a perfect square. We say that A ^c =[c _ij ] is the conjugate adjacency matrix of a graph G if c _ij =c for any two adjacent vertices i and j, $c_{ij}=\overline{c}$ for any two nonadjacent vertices i and j, and c _ij =0 if i=j. Let P _G ^c (λ)=|λ I?A ^c | denote the conjugate characteristic polynomial of G. Further, let e=e(G) and Δ=Δ(G) be the number of edges and number of triangles of G, respectively. Let G and H be two graphs of order n and let e(G)=e(H). In this work we prove that c ₃(G)=c ₃(H) if and only if Δ(G)=Δ(H) and $\Delta(\overline{G})=\Delta(\overline{H})$ , where $\overline{G}$ denotes the complement of G and c _k is the coefficient which corresponds to λ ^n?k with respect to P _G ^c (λ). Besides, we here give the conjugate spectrum and conjugate characteristic polynomial of all connected graphs of order n=2,3,4,5, with respect to the constant $c=1+\sqrt{2}$ .     

On Group Choosability of Graphs,II

Given a group A and a directed graph G, let F(G, A) denote the set of all maps ${f : E(G) \rightarrow A}$ . Fix an orientation of G and a list assignment ${L : V(G) \mapsto 2^A}$ . For an ${f \in F(G, A)}$ , G is (A, L, f)-colorable if there exists a map ${c:V(G) \mapsto \cup_{v \in V(G)}L(v)}$ such that ${c(v) \in L(v)}$ , ${\forall v \in V(G)}$ and ${c(x)-c(y)\neq f(xy)}$ for every edge e = xy directed from x to y. If for any ${f\in F(G,A)}$ , G has an (A, L, f)-coloring, then G is (A, L)-colorable. If G is (A, L)-colorable for any group A of order at least k and for any k-list assignment ${L:V(G) \rightarrow 2^A}$ , then G is k-group choosable. The group choice number, denoted by ${\chi_{gl}(G)}$ , is the minimum k such that G is k-group choosable. In this paper, we prove that every planar graph is 5-group choosable, and every planar graph with girth at least 5 is 3-group choosable. We also consider extensions of these results to graphs that do not have a K ₅ or a K _3,3 as a minor, and discuss group choosability versions of Hadwiger’s and Woodall’s conjectures.     

On a conjecture about automorphisms of finite p-groups

Let G be a nonabelian finite p-group. A longstanding conjecture asserts that G admits a noninner automorphism of order p. In this paper, we prove that if G satisfies one of the following conditions

1. ${\mathrm{rank}(G'\cap Z(G))\neq \mathrm{rank}(Z(G))}$
2. ${\frac{Z_{2}(G)}{Z(G)}}$ is cyclic
3. C _G (Z(Φ(G))) = Φ(G) and ${\frac{Z_{2}(G)\cap Z(\Phi(G))}{Z(G)} }$ is not elementary abelian of rank rs, where r = d(G) and s = rank (Z(G)),

then G has a noninner central automorphism of order p which fixes Φ(G) elementwise.     

Contractible Edges in k-Connected Graphs with Some Forbidden Subgraphs

In 2001, Kawarabayashi proved that for any odd integer k ≥ 3, if a k-connected graph G is \({K^{-}_{4}}\) -free, then G has a k-contractible edge. He pointed out, by a counterexample, that this result does not hold when k is even. In this paper, we have proved the following two results on the subject: (1) For any even integer k ≥ 4, if a k-connected graph G is \({K_{4}^{-}}\) -free and d _G (x) + d _G (y) ≥ 2k + 1 hold for every two adjacent vertices x and y of V(G), then G has a k-contractible edge. (2) Let t ≥ 3, k ≥ 2t – 1 be integers. If a k-connected graph G is \({(K_{1}+(K_{2} \cup K_{1, t}))}\) -free and d _G (x) + d _G (y) ≥ 2k + 1 hold for every two adjacent vertices x and y of V(G), then G has a k-contractible edge.     

Affine groups and flag-transitive triplanes

Let D be a nontrivial 2-(v, k, 3) symmetric design (triplane) and let G≤Aut(D) be flag-transitive and point-primitive. In this paper, we prove that if G is an affine group, then G≤AΓL1(q), where q is some power of a prime p and p≥5.     

Nowhere-Zero 5-Flows and Even (1,2)-Factors

A graph G = (V, E) admits a nowhere-zero k-flow if there exists an orientation H = (V, A) of G and an integer flow ${\varphi:A \to \mathbb{Z}}$ such that for all ${a \in A, 0 < |\varphi(a)| < k}$ . Tutte conjectured that every bridgeless graphs admits a nowhere-zero 5-flow. A (1,2)-factor of G is a set ${F \subseteq E}$ such that the degree of any vertex v in the subgraph induced by F is 1 or 2. Let us call an edge of G, F-balanced if either it belongs to F or both its ends have the same degree in F. Call a cycle of G F-even if it has an even number of F-balanced edges. A (1,2)-factor F of G is even if each cycle of G is F-even. The main result of the paper is that a cubic graph G admits a nowhere-zero 5-flow if and only if G has an even (1,2)-factor.     

Rainbow Connection Number, Bridges and Radius

Let G be a connected graph. The notion of rainbow connection number rc(G) of a graph G was introduced by Chartrand et al. (Math Bohem 133:85–98, 2008). Basavaraju et al. (arXiv:1011.0620v1 [math.CO], 2010) proved that for every bridgeless graph G with radius r, ${rc(G)\leq r(r+2)}$ and the bound is tight. In this paper, we show that for a connected graph G with radius r and center vertex u, if we let D ^r  = {u}, then G has r?1 connected dominating sets ${ D^{r-1}, D^{r-2},\ldots, D^{1}}$ such that ${D^{r} \subset D^{r-1} \subset D^{r-2} \cdots\subset D^{1} \subset D^{0}=V(G)}$ and ${rc(G)\leq \sum_{i=1}^{r} \max \{2i+1,b_i\}}$ , where b _i is the number of bridges in E[D ⁱ , N(D ⁱ )] for ${1\leq i \leq r}$ . From the result, we can get that if ${b_i\leq 2i+1}$ for all ${1\leq i\leq r}$ , then ${rc(G)\leq \sum_{i=1}^{r}(2i+1)= r(r+2)}$ ; if b _i  > 2i + 1 for all ${1\leq i\leq r}$ , then ${rc(G)= \sum_{i=1}^{r}b_i}$ , the number of bridges of G. This generalizes the result of Basavaraju et al. In addition, an example is given to show that there exist infinitely graphs with bridges whose rc(G) is only dependent on the radius of G, and another example is given to show that there exist infinitely graphs with bridges whose rc(G) is only dependent on the number of bridges in G.     

A Spanning Tree with High Degree Vertices

Let G be a connected graph, let ${X \subset V(G)}$ and let f be a mapping from X to {2, 3, . . .}. Kaneko and Yoshimoto (Inf Process Lett 73:163–165, 2000) conjectured that if |N _G (S) ? X| ≥ f (S) ? 2|S| + ω _G (S) + 1 for any subset ${S \subset X}$ , then there exists a spanning tree T such that d _T (x) ≥ f (x) for all ${x \in X}$ . In this paper, we show a result with a stronger assumption than this conjecture; if |N _G (S) ? X| ≥ f (S) ? 2|S| + α(S) + 1 for any subset ${S \subset X}$ , then there exists a spanning tree T such that d _T (x) ≥ f (x) for all ${x \in X}$ .     

Bounds on the 2-Rainbow Domination Number of Graphs

A 2-rainbow domination function of a graph G is a function f that assigns to each vertex a set of colors chosen from the set {1, 2}, such that for any ${v\in V(G), f(v)=\emptyset}$ implies ${\bigcup_{u\in N(v)}f(u)=\{1,2\}.}$ The 2-rainbow domination number γ _r2(G) of a graph G is the minimum ${w(f)=\Sigma_{v\in V}|f(v)|}$ over all such functions f. Let G be a connected graph of order |V(G)| = n ≥ 3. We prove that γ _r2(G) ≤ 3n/4 and we characterize the graphs achieving equality. We also prove a lower bound for 2-rainbow domination number of a tree using its domination number. Some other lower and upper bounds of γ _r2(G) in terms of diameter are also given.     

Centralizers of coprime automorphisms of finite groups

Let A be an elementary abelian group of order p ^k with k ≥ 3 acting on a finite p′-group G. The following results are proved. If γ _k-2(C _G (a)) is nilpotent of class at most c for any ${a \in A^{\#}}$ , then γ _k-2(G) is nilpotent and has {c, k, p}-bounded nilpotency class. If, for some integer d such that 2 ^d  + 2 ≤ k, the dth derived group of C _G (a) is nilpotent of class at most c for any ${a \in A^{\#}}$ , then the dth derived group G ^(d) is nilpotent and has {c, k, p}-bounded nilpotency class.     

A special f-edge cover-coloring of multigraphs

An edge cover-coloring of G is called a special (f,g)-edge cover-coloring, if each color appears at each vertex at least f(v) times and the number of multiple edges receive the same color is at most g(vw) for vw∈E(G). Let $\chi_{f_{g}}''$ denote the maximum positive integer k for which using k colors a special (f,g)-edge cover-coloring of G exists. The existence of $\chi_{f_{g}}''$ is discussed and the lower bound of $\chi_{f_{g}}''$ is obtained.     

Comparison of topologies on ⁎-algebras of locally measurable operators

We consider the local measure topology ${t(\mathcal{M})}$ on the ?-algebra ${LS(\mathcal{M})}$ of all locally measurable operators and on the ?-algebra ${S(\mathcal{M},\tau)}$ of all τ-measurable operators affiliated with a von Neumann algebra ${\mathcal{M}}$ . If τ is a semifinite but not a finite trace on ${\mathcal{M},}$ then one can consider the τ-local measure topology t _τ l and the weak τ-local measure topology t _{w τ l} . We study relationships between the topology ${t(\mathcal{M})}$ and the topologies t _τ l , t _{w τ l} , and the (o)-topology ${t_o(\mathcal{M})}$ on ${LS_h(\mathcal{M})=\{T\in LS(\mathcal{M}): T^\ast=T\}}$ . We find that the topologies ${t(\mathcal{M})}$ and t _τ l (resp. ${t(\mathcal{M})}$ and t _{w τ l} ) coincide on ${S(\mathcal{M},\tau)}$ if and only if ${\mathcal{M}}$ is finite, and ${t(\mathcal{M})=t_o(\mathcal{M})}$ on ${LS_h(\mathcal{M})}$ holds if and only if ${\mathcal{M}}$ is a σ-finite and finite. Moreover, it turns out that the topology t _{τ l} (resp. t _{w τ l} ) coincides with the (o)-topology on ${S_h(\mathcal{M},\tau)}$ only for finite traces. We give necessary and sufficient conditions for the topology ${t(\mathcal{M})}$ to be locally convex (resp., normable). We show that (o)-convergence of sequences in ${LS_h(\mathcal{M})}$ and convergence in the topology ${t(\mathcal{M})}$ coincide if and only if the algebra ${\mathcal{M}}$ is an atomic and finite algebra.     

Linear 2-Arboricity of Planar Graphs with Neither 3-Cycles Nor Adjacent 4-Cycles

Let G be a planar graph with neither 3-cycles nor adjacent 4-cycles. We prove that if G is connected and δ(G) ≥ 2, then G contains an edge uv with d(u) + d(v) ≤ 7 or a 2-alternating cycle. By this result, we obtain that G’s linear 2-arboricity ${la_{2}(G)\leq\lceil\frac{\Delta(G)+1}{2}\rceil+4.}$ .     

Group Connectivity of Bridged Graphs

Let G be a 2-edge-connected simple graph, and let A denote an abelian group with the identity element 0. If a graph G ^* is obtained by repeatedly contracting nontrivial A-connected subgraphs of G until no such a subgraph left, we say G can be A-reduced to G*. A graph G is bridged if every cycle of length at least 4 has two vertices x, y such that d _G (x, y) < d _C (x, y). In this paper, we investigate the group connectivity number Λ _g (G) = min{n: G is A-connected for every abelian group with |A| ≥ n} for bridged graphs. Our results extend the early theorems for chordal graphs by Lai (Graphs Comb 16:165–176, 2000) and Chen et al. (Ars Comb 88:217–227, 2008).     

Polaroid Operators with SVEP and Perturbations of Property (gw)

Let ${\mathcal{L}(X)}$ be the algebra of all bounded linear operators on X and ${\mathcal{P}S(X)}$ be the class of polaroid operators with the single-valued extension property. The property (gw) holds for ${T \in \mathcal{L}(X)}$ if the complement in the approximate point spectrum of the semi-B-essential approximate point spectrum coincides with the set of all isolated points of the spectrum which are eigenvalues of the spectrum. In this note we focus on the stability of the property (gw) under perturbations: we prove that, if ${T \in \mathcal{P}S(X)}$ and A (resp. Q) is an algebraic (resp. quasinilpotent) operator, then the property (gw) holds for f(T ^* + A ^*) (resp. f(T ^* + Q*)) for every analytic function f in σ(T + A) (resp. σ(T + Q)). Some applications are also given.     

Minimal logarithmic signatures for finite groups of Lie type

A logarithmic signature (LS) for a finite group G is an ordered tuple α =  [A ₁, A ₂, . . . , A _n ] of subsets A _i of G, such that every element ${g \in G}$ can be expressed uniquely as a product g =  a ₁ a ₂ . . . a _n , where ${a_i \in A_i}$ . The length of an LS α is defined to be ${l(\alpha)= \sum^{n}_{i=1}|A_i|}$ . It can be easily seen that for a group G of order ${\prod^k_{j=1}{p_j}^{m_j}}$ , the length of any LS α for G, satisfies, ${l(\alpha) \geq \sum^k_{j=1}m_jp_j}$ . An LS for which this lower bound is achieved is called a minimal logarithmic signature (MLS) (González Vasco et al., Tatra Mt. Math. Publ. 25:2337, 2002). The MLS conjecture states that every finite simple group has an MLS. This paper addresses the MLS conjecture for classical groups of Lie type and is shown to be true for the families PSL _n (q) and PSp _2n (q). Our methods use Singer subgroups and the Levi decomposition of parabolic subgroups for these groups.