Total colorings of graphs of order 2n having maximum degree 2n−2

Let χ _t (G) and †(G) denote respectively the total chromatic number and maximum degree of graphG. Yap, Wang and Zhang proved in 1989 that ifG is a graph of orderp having †(G)≥p−4, then χ _t (G≤Δ(G)+2. Hilton has characterized the class of graphG of order 2n having †(G)=2n−1 such that χ _t (G=Δ(G)+2. In this paper, we characterize the class of graphsG of order 2n having †(G)=2n−2 such that χ _t (G=Δ(G)+2 Research supported by National Science Council of the Republic of China (NSC 79-0208-M009-15)     

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.     

On the total domination subdivision numbers in graphs

A set S of vertices of a graph G = (V, E) without isolated vertex is a total dominating set if every vertex of V(G) is adjacent to some vertex in S. The total domination number γ _t (G) is the minimum cardinality of a total dominating set of G. The total domination subdivision number sd_γt (G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the total domination number. Karami, Khoeilar, Sheikholeslami and Khodkar, (Graphs and Combinatorics, 2009, 25, 727–733) proved that for any connected graph G of order n ≥ 3, sd_{γ t} (G) ≤ 2γ _t (G) − 1 and posed the following problem: Characterize the graphs that achieve the aforementioned upper bound. In this paper we first prove that sd_{γ t} (G) ≤ 2α′(G) for every connected graph G of order n ≥ 3 and δ(G) ≥ 2 where α′(G) is the maximum number of edges in a matching in G and then we characterize all connected graphs G with sd_{γ t} (G)=2γ _t (G)−1.     

Permutability of minimal subgroups and<Emphasis Type="Italic">p</Emphasis>-nilpotency of finite groups

In this paper it is proved that ifp is a prime dividing the order of a groupG with (|G|,p − 1) = 1 andP a Sylowp-subgroup ofG, thenG isp-nilpotent if every subgroup ofP ∩G ^N of orderp is permutable inN _G (P) and whenp = 2 either every cyclic subgroup ofP ∩G ^N of order 4 is permutable inN _G (P) orP is quaternion-free. Some applications of this result are given. The research of the first author is supported by a grant of Shanxi University and a research grant of Shanxi Province, PR China. The research of the second author is partially supported by a UGC(HK) grant #2160126 (1999/2000).     

Linearity of Zp[[t]]-Perfect Groups

Let G be a _p[[t]]-standard group of level 1. Then G is _p[[t]]-perfect if its lower central series is given by powers of the maximal ideal (p, t), i.e. if _n(G) = G((p,t)ⁿ). We prove that a _p[[t]]-perfect group is linear by imitating the proof that a _p[[t]]-standard group is linear.     

The size-Ramsey number of trees

IfG andH are graphs, let us writeG→(H)₂ ifG contains a monochromatic copy ofH in any 2-colouring of the edges ofG. Thesize-Ramsey number r _e(H) of a graphH is the smallest possible number of edges a graphG may have ifG→(H)₂. SupposeT is a tree of order |T|≥2, and lett ₀,t ₁ be the cardinalities of the vertex classes ofT as a bipartite graph, and let Δ(T) be the maximal degree ofT. Moreover, let Δ₀, Δ₁ be the maxima of the degrees of the vertices in the respective vertex classes, and letβ(T)=T ₀Δ₀+t ₁Δ₁. Beck [7] proved thatβ(T)/4≤r _e(T)=O{β(T)(log|T|)¹²}, improving on a previous result of his [6] stating thatr _e(T)≤Δ(T)|T|(log|T|)¹². In [6], Beck conjectures thatr _e(T)=O{Δ(T)|T|}, and in [7] he puts forward the stronger conjecture thatr _e(T)=O{β(T)}. Here, we prove the first of these conjectures, and come quite close to proving the second by showing thatr _e(T)=O{β(T)logΔ(T)}.     

Isoperimetric inequalities in potential theory

Given a non-empty bounded domainG in ⁿ ,n2, letr ₀(G) denote the radius of the ballG ₀ having center 0 and the same volume asG. The exterior deficiencyd _e (G) is defined byd _e (G)=r _e (G)/r ₀(G)–1 wherer _e (G) denotes the circumradius ofG. Similarlyd _i (G)=1–r _i (G)/r ₀(G) wherer _i (G) is the inradius ofG. Various isoperimetric inequalities for the capacity and the first eigenvalue ofG are shown. The main results are of the form CapG(1+cf(d _e (G)))CapG ₀ and ₁(G)(1+cf(d _i (G)))₁(G ₀),f(t)=t ³ ifn=2,f(t)=t ³/(ln 1/t) ifn=3,f(t)=t ^(n+3)/2 ifn4 (for convex G and small deficiencies ifn3).     

A note on generalized characters

For a compactly generated LCA group G, it is shown that the setH(G) of all generalized characters on G equipped with the compact-open topology is a LCA group andH(G) = Ĝ (the dual group ofG) if and only ifG is compact. Both results fail for arbitrary LCA groups. Further, ifG is second countable, then the Gel’fand space of the commutative convolution algebraC _c (G) equipped with the inductive limit topology is topologically homeomorphic toH(G).     

On r-recognition by prime graph of Bp(3) where p is an odd prime

Let G be a finite group. The prime graph Γ(G) of G is defined as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p and p′ are joined by an edge if there is an element in G of order pp′. We denote by k(Γ(G)) the number of isomorphism classes of finite groups H satisfying Γ(G) = Γ(H). Given a natural number r, a finite group G is called r-recognizable by prime graph if k(Γ(G)) =  r. In Shen et al. (Sib. Math. J. 51(2):244–254, 2010), it is proved that if p is an odd prime, then B _p (3) is recognizable by element orders. In this paper as the main result, we show that if G is a finite group such that Γ(G) = Γ(B _p (3)), where p > 3 is an odd prime, then G @ B_p(3){G\cong B_p(3)} or C _p (3). Also if Γ(G) = Γ(B ₃(3)), then G @ B₃(3), C₃(3), D₄(3){G\cong B_3(3), C_3(3), D_4(3)}, or G/O₂(G) @ Aut(²B₂(8)){G/O_2(G)\cong {\rm Aut}(^2B_2(8))}. As a corollary, the main result of the above paper is obtained.     

Path extendable graphs

A pathP in a graphG is said to beextendable if there exists a pathP’ inG with the same endvertices asP such thatV(P)⊆V (P’) and |V(P’)|=|V(P)|+1. A graphG ispath extendable if every nonhamiltonian path inG is extendable. We investigate the extent to which known sufficient conditions for a graph to be hamiltonian-connected imply the extendability of paths in the graph. Several theorems are proved: for example, it is shown that ifG is a graph of orderp in which the degree sum of each pair of non-adjacent vertices is at leastp+1 andP is a nonextendable path of orderk inG thenk≤(p+1)/2 and 〈V (P)〉≅K _k orK _k −e. As corollaries of this we deduce that if δ(G)≥(p+2)/2 or if the degree sum of each pair of nonadjacent vertices inG is at least (3p−3)/2 thenG is path extendable, which strengthen results of Williamson [13].     

An Upper Bound for the Total Domination Subdivision Number of a Graph

A set S of vertices of a graph G = (V, E) without isolated vertex is a total dominating set if every vertex of V(G) is adjacent to some vertex in S. The total domination number γ _t (G) is the minimum cardinality of a total dominating set of G. The total domination subdivision number sd_gt(G){{\rm sd}_{\gamma_t}(G)} is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the total domination number. In this paper, we prove that sd_gt(G) ￡ 2g_t(G)-1{{\rm sd}_{\gamma_t}(G)\leq 2\gamma_t(G)-1} for every simple connected graph G of order n ≥ 3.     

Minimal presentations for certain group extensions

IfG is a finite group thend(G) denotes the minimal number of generators ofG. IfH andK are groups then the extension, 1 →H →G →K → 1, is called an outer extension ofK byH ifd(G)=d(H)+d(K). Let be the class of groups containing all finitep-groupsG which has a presentation withd(G) = dimH ¹(G,z _p ) generators andr(G)=dimH ² (G,Z _p ) relations: in this article it is shown that ifK is a non cyclic group belonging to andH is a finite abelian p-group then any outer extension ofK byH belongs to .     

On a Function Related to Multinomial Coefficients (I)

Let F _p,t (n) denote the number of the coefficients of (x ₁+1x ₂+...+x _t ) ^j , 0 ≤j≤n− 1, which are not divisible by the prime p. Define G _p,t (n) = F _p,t /n ^θ and β(p,t) = lim infF _p,t )(n)/n ^θ, where θ = (log)/(log p). In this paper, we mainly prove that G _p,t can be extended to a continuous function on ℝ⁺, and the function G _p,t is nowhere monotonic. Both the set of differential points of the function G _p,t and the set of non-differential points of the function G _p,t are dense in ℝ⁺. Received February 18, 2000, Accepted December 7, 2000     

On the multiplicity of the eigenvalues of a graph

Given a graph G with characteristic polynomial ϕ(t), we consider the ML-decomposition ϕ(t) = q ₁(t)q ₂(t)² ... q _m (t)^m, where each q _i (t) is an integral polynomial and the roots of ϕ(t) with multiplicity j are exactly the roots of q _j (t). We give an algorithm to construct the polynomials q _i (t) and describe some relations of their coefficients with other combinatorial invariants of G. In particular, we get new bounds for the energy E(G) = |λ_i| of G, where λ₁, λ₂, ..., λ_n are the eigenvalues of G (with multiplicity). Most of the results are proved for the more general situation of a Hermitian matrix whose characteristic polynomial has integral coefficients. This work was done during a visit of the second named author to UNAM.     

The co-degrees of irreducible characters

LetG be a finite group. The co-degree of an irreducible character χ ofG is defined to be the number |G|/χ(1). The set of all prime divisors of all the co-degrees of the nonlinear irreducible characters ofG is denoted by Σ(G). First we show that Σ(G)=π(G) (the set of all prime divisors of |G|) unlessG is nilpotent-by-abelian. Then we make Σ(G) a graph by adjoining two elements of Σ(G) if and only if their product divides a co-degree of some nonlinear character ofG. We show that the graph Σ(G) is connected and has diameter at most 2. Additional information on the graph is given. These results are analogs to theorems obtained for the graph corresponding to the character degrees (by Manz, Staszewski, Willems and Wolf) and for the graph corresponding to the class sizes (by Bertram, Herzog and Mann). Finally, we investigate groups with some restriction on the co-degrees. Among other results we show that ifG has a co-degree which is ap-power for some primep, then the corresponding character is monomial andO _p (G)≠1. Also we describe groups in which each co-degree of a nonlinear character is divisible by at most two primes. These results generalize results of Chillag and Herzog. Other results are proved as well. The paper was written during this author’s visit at the Technion and the University of Tel Aviv. He would like to thank the departments of mathematics at the Technion and the University of Tel Aviv for their hospitality and support.     

Longest Paths and Longest Cycles in Graphs with Large Degree Sums

 Let p(G) and c(G) denote the number of vertices in a longest path and a longest cycle, respectively, of a finite, simple graph G. Define σ₄(G)=min{d(x ₁)+d(x ₂)+ d(x ₃)+d(x ₄) | {x ₁,…,x ₄} is independent in G}. In this paper, the difference p(G)−c(G) is considered for 2-connected graphs G with σ₄(G)≥|V(G)|+3. Among others, we show that p(G)−c(G)≤2 or every longest path in G is a dominating path. Received: August 28, 2000 Final version received: May 23, 2002     

Groups with few characters of small degrees

Letd>1 be a proper divisor of the order of a finite groupG and let σ _d (G) be the sum of squares of degrees of those irreducible characters whose degrees are not divisible byd. It is easy to see thatd divides σ _d (G). The groupsG such that σ _d (G) =d coincide with Frobenius groups whose kernel has indexd (see G. Karpilovsky,Group Representations, Volume 1, Part B, North-Holland, Amsterdam, 1992, Theorem 37.5.5). In this note we study the case σ _d (G) = 2d in some detail. In particular, ifG is a 2-group, it is of maximal class (Remark 3(b)). The author was supported in part by the Ministry of Absorption of Israel.     

The character table of the group GL 2(Q) when extended by a certain group of order two

LetG denote either of the groupsGL ₂(q) or SL₂(q). Then θ :G →G given by θ(A) = (A ^t)^t, whereA ^t denotes the transpose of the matrixA, is an automorphism ofG. Therefore we may form the groupG.θ> which is the split extension of the groupG by the cyclic group θ of order 2. Our aim in this paper is to find the complex irreducible character table ofG. θ.     

On a class of groups of prime-power order

LetG be a finitep-group,d(G)=dimH ¹ (G, Z _p) andr(G)=dimH ²(G, Z_p). Thend(G) is the minimal number of generators ofG, and we say thatG is a member of a classG _p of finitep-groups ifG has a presentation withd(G) generators andr(G) relations. We show that ifG is any finitep-group, thenG is the direct factor of a member ofG _p by a member ofG _p .     

Relative Length of Longest Paths and Cycles in Graphs

For a graph G, let diff(G) = p(G) − c(G), where p(G) and c(G) denote the orders of a longest path and a longest cycle in G, respectively. Let G be a 3-connected graph of order n. In the paper, we give a best-possible lower bound to σ₄(G) to assure diff(G) ≤ 1. The result settles a conjecture in J. Graph Theory 37 (2001), 137–156.