Multicoloring and Mycielski construction

The generalized Mycielskians of graphs (also known as cones over graphs) are the natural generalization of the Mycielskians of graphs (which were first introduced by Mycielski in 1955). Given a graph G and any integer p?0, one can transform G into a new graph μ_p(G), the p-Mycielskian of G. In this paper, we study the kth chromatic numbers χ_k of Mycielskians and generalized Mycielskians of graphs. We show that χ_k(G)+1?χ_k(μ(G))?χ_k(G)+k, where both upper and lower bounds are attainable. We then investigate the kth chromatic number of Mycielskians of cycles and determine the kth chromatic number of p-Mycielskian of a complete graph K_n for any integers k?1, p?0 and n?2. Finally, we prove that if a graph G is a/b-colorable then the p-Mycielskian of G, μ_p(G), is (at+b^p+1)/bt-colorable, where . And thus obtain graphs G with m(G) grows exponentially with the order of G, where m(G) is the minimal denominator of a a/b-coloring of G with χ_f(G)=a/b.     

The boundary of the outer space of a free product

Let G be a countable group that splits as a free product of groups of the form G = G ₁ *···* G _k * F _N , where F _N is a finitely generated free group. We identify the closure of the outer space PO(G, {G ₁,..., G _k }) for the axes topology with the space of projective minimal, very small (G, {G ₁,..., G _k })-trees, i.e. trees whose arc stabilizers are either trivial, or cyclic, closed under taking roots, and not conjugate into any of the G _i ’s, and whose tripod stabilizers are trivial. Its topological dimension is equal to 3N + 2k ? 4, and the boundary has dimension 3N + 2k ? 5. We also prove that any very small (G, {G ₁,..., G _k })-tree has at most 2N + 2k?2 orbits of branch points.     

The multiple originator broadcasting problem in graphs

Given a graph G and a vertex subset S of V(G), the broadcasting time with respect toS, denoted by b(G,S), is the minimum broadcasting time when using S as the broadcasting set. And the k-broadcasting number, denoted by b_k(G), is defined by b_k(G)=min{b(G,S)|S⊆V(G),|S|=k}.Given a graph G and two vertex subsets S, S^′ of V(G), define , d(S,S^′)=min{d(u,v)|u∈S, v∈S^′}, and for all v∈V(G). For all k, 1?k?|V(G)|, the k-radius of G, denoted by r_k(G), is defined as r_k(G)=min{d(G,S)|S⊆V(G), |S|=k}.In this paper, we study the relation between the k-radius and the k-broadcasting numbers of graphs. We also give the 2-radius and the 2-broadcasting numbers of the grid graphs, and the k-broadcasting numbers of the complete n-partite graphs and the hypercubes.     

Lower bounds on the signed total k-domination number of graphs

Let G be a graph with vertex set V(G). For any integer k ≥ 1, a signed total k-dominating function is a function f: V(G) → {?1, 1} satisfying ∑_x∈N(v)f(x) ≥ k for every v ∈ V(G), where N(v) is the neighborhood of v. The minimum of the values ∑_v∈V(G)f(v), taken over all signed total k-dominating functions f, is called the signed total k-domination number. In this note we present some new sharp lower bounds on the signed total k-domination number of a graph. Some of our results improve known bounds.     

Frattini and related subgroups of mapping class groups

Let Γ_g,b denote the orientation-preserving mapping class group of a closed orientable surface of genus g with b punctures. For a group G let Φ_f(G) denote the intersection of all maximal subgroups of finite index in G. Motivated by a question of Ivanov as to whether Φ_f(G) is nilpotent when G is a finitely generated subgroup of Γ_g,b, in this paper we compute Φ_f(G) for certain subgroups of Γ_g,b. In particular, we answer Ivanov’s question in the affirmative for these subgroups of Γg,b.     

Seperating the intrinsic complexity and the derivational complexity of the word problem for finitely presented groups

A pseudo-natural algorithm for the word problem of a finitely presented group is an algorithm which not only tells us whether or not a word w equals 1 in the group but also gives a derivation of 1 from w when w equals 1. In [13], [14] Madlener and Otto show that, if we measure complexity of a primitive recursive algorithm by its level in the Grzegorczyk hierarchy, there are groups in which a pseudo-natural algorithm is arbitrarily more complicated than an algorithm which simply solves the word problem. In a given group the lowest degree of complexity that can be realised by a pseudo-natural algorithm is essentially the derivational complexity of that group. Thus the result separates the derivational complexity of the word problem of a finitely presented group from its intrinsic complexity. The proof given in [13] involves the construction of a finitely presented group G from a Turing machine T such that the intrinsic complexity of the word problem for G reflects the complexity of the halting problem of T, while the derivational complexity of the word problem for G reflects the runtime complexity of T. The proof of one of the crucial lemmas in [13] is only sketched, and part of the purpose of this paper is to give the full details of this proof. We will also obtain a variant of their proof, using modular machines rather than Turing machines. As for several other results, this simplifies the proofs considerably. MSC: 03D40, 20F10.     

Rainbow domination on trees

This paper studies a variation of domination in graphs called rainbow domination. For a positive integer k, a k-rainbow dominating function of a graph G is a function f from V(G) to the set of all subsets of {1,2,…,k} such that for any vertex v with f(v)=0? we have ∪_u∈NG(v)f(u)={1,2,…,k}. The 1-rainbow domination is the same as the ordinary domination. The k-rainbow domination problem is to determine the k-rainbow domination number of a graph G, that is the minimum value of ∑_v∈V(G)|f(v)| where f runs over all k-rainbow dominating functions of G. In this paper, we prove that the k-rainbow domination problem is NP-complete even when restricted to chordal graphs or bipartite graphs. We then give a linear-time algorithm for the k-rainbow domination problem on trees. For a given tree T, we also determine the smallest k such that .     

Φ-Harmonic functions on discrete groups and the first ℓΦ-cohomology

We study the first cohomology groups of a countable discrete group G with coefficients in a G-module ?^Φ(G), where Φ is an N-function of class Δ₂(0) ∩ ?₂(0). Developing the ideas of Puls and Martin-Valette for a finitely generated group G, we introduce the discrete Φ-Laplacian and prove a theorem on the decomposition of the space of Φ-Dirichlet finite functions into the direct sum of the spaces of Φ-harmonic functions and ?^Φ(G) (with an appropriate factorization). We prove also that if a finitely generated group G has a finitely generated infinite amenable subgroup with infinite centralizer then \(\bar H^1\) (G, ?^Φ(G)) = 0. In conclusion, we show the triviality of the first cohomology group for the wreath product of two groups one of which is nonamenable.     

Chromatic capacity and graph operations

The chromatic capacityχ_cap(G) of a graph G is the largest k for which there exists a k-coloring of the edges of G such that, for every coloring of the vertices of G with the same colors, some edge is colored the same as both its vertices. We prove that there is an unbounded function f:N→N such that χ_cap(G)?f(χ(G)) for almost every graph G, where χ denotes the chromatic number. We show that for any positive integers n and k with k?n/2 there exists a graph G with χ(G)=n and χ_cap(G)=n-k, extending a result of Greene. We obtain bounds on that are tight as r→∞, where is the complete n-partite graph with r vertices in each part. Finally, for any positive integers p and q we construct a graph G with χ_cap(G)+1=χ(G)=p that contains no odd cycles of length less than q.     

Signed k-independence in graphs

Let k ≥ 2 be an integer. A function f: V(G) → {?1, 1} defined on the vertex set V(G) of a graph G is a signed k-independence function if the sum of its function values over any closed neighborhood is at most k ? 1. That is, Σ _x∈N[v] f(x) ≤ k ? 1 for every v ∈ V(G), where N[v] consists of v and every vertex adjacent to v. The weight of a signed k-independence function f is w(f) = Σ _v∈V(G) f(v). The maximum weight w(f), taken over all signed k-independence functions f on G, is the signed k-independence number α _s ^k (G) of G. In this work, we mainly present upper bounds on α _s ^k (G), as for example α _s ^k (G) ≤ n ? 2?(Δ(G) + 2 ? k)/2?, and we prove the Nordhaus-Gaddum type inequality $\alpha _S^k \left( G \right) + \alpha _S^k \left( {\bar G} \right) \leqslant n + 2k - 3$ , where n is the order, Δ(G) the maximum degree and $\bar G$ the complement of the graph G. Some of our results imply well-known bounds on the signed 2-independence number.     

Variations of Landau’s theorem for p-regular and p-singular conjugacy classes

The well-known Landau’s theorem states that, for any positive integer k, there are finitely many isomorphism classes of finite groups with exactly k (conjugacy) classes. We study variations of this theorem for p-regular classes as well as p-singular classes. We prove several results showing that the structure of a finite group is strongly restricted by the number of p-regular classes or the number of p-singular classes of the group. In particular, if G is a finite group with O_p(G) = 1 then |G/F(G)|_p' is bounded in terms of the number of p-regular classes of G. However, it is not possible to prove that there are finitely many groups with no nontrivial normal p-subgroup and kp-regular classes without solving some extremely difficult number-theoretic problems (for instance, we would need to show that the number of Fermat primes is finite).     

On the Characterizability of the Automorphism Groups of Sporadic Simple Groups by Their Element Orders

For G a finite group, π _e (G) denotes the set of orders of elements in G. If Ω is a subset of the set of natural numbers, h(Ω) stands for the number of isomorphism classes of finite groups with the same set Ω of element orders. We say that G is k-distinguishable if h(π _e (G)) = k < ∞, otherwise G is called non-distinguishable. Usually, a 1-distinguishable group is called a characterizable group. It is shown that if M is a sporadic simple group different from M ₁₂, M ₂₂, J ₂, He, Suz, M ^c L and O′N, then Aut(M) is characterizable by its element orders. It is also proved that if M is isomorphic to M ₁₂, M ₂₂, He, Suz or O′N, then h(π _e (Aut(M))) ∈¸ {1,∞}.     

The {k}-domatic number of a graph

For a positive integer k, a {k}-dominating function of a graph G is a function f from the vertex set V(G) to the set {0, 1, 2, . . . , k} such that for any vertex ${v\in V(G)}$ , the condition ${\sum_{u\in N[v]}f(u)\ge k}$ is fulfilled, where N[v] is the closed neighborhood of v. A {1}-dominating function is the same as ordinary domination. A set {f ₁, f ₂, . . . , f _d } of {k}-dominating functions on G with the property that ${\sum_{i=1}^df_i(v)\le k}$ for each ${v\in V(G)}$ , is called a {k}-dominating family (of functions) on G. The maximum number of functions in a {k}-dominating family on G is the {k}-domatic number of G, denoted by d _{k}(G). Note that d _{1}(G) is the classical domatic number d(G). In this paper we initiate the study of the {k}-domatic number in graphs and we present some bounds for d _{k}(G). Many of the known bounds of d(G) are immediate consequences of our results.     

Some Bistar Bipartite Ramsey Numbers

For bipartite graphs G ₁, G ₂, . . . ,G _k , the bipartite Ramsey number b(G ₁, G ₂, . . . , G _k ) is the least positive integer b so that any colouring of the edges of K _b,b with k colours will result in a copy of G _i in the ith colour for some i. A tree of diameter three is called a bistar, and will be denoted by B(s, t), where s ≥ 2 and t ≥ 2 are the degrees of the two support vertices. In this paper we will obtain some exact values for b(B(s, t), B(s, t)) and b(B(s, s), B(s, s)). Furtermore, we will show that if k colours are used, with k ≥ 2 and s ≥ 2, then \({b_{k}(B(s, s)) \leq \lceil k(s - 1) + \sqrt{(s - 1)^{2}(k^{2} - k) - k(2s - 4)} \rceil}\) . Finally, we show that for s ≥ 3 and k ≥ 2, the Ramsey number \({r_{k}(B(s, s)) \leq \lceil 2k(s - 1)+ \frac{1}{2} + \frac{1}{2} \sqrt{(4k(s - 1) + 1)^{2} - 8k(2s^{2} - s - 2)} \rceil}\) .     

On Groups Generated by Elements of Pirme Order

In the first part of the paper we give a characterization of groups generated by elements of fixed prime order p. In the second part we study the group G _n ^(p) of n × n matrices with the pth power of the determinant equal to 1 over a field F containing a primitive pth root of 1. It is known that the group G _n ⁽²⁾ of n × n matrices of determinant ± 1 over a field F and the group SL _n (F) are generated by their involutions and that each element in these groups is a product of four involutions. We consider some subgroups G of G _n ^(p) and study the following problems: Is G generated by its elements of order p? If so, is every element of G a product of k elements of order p for some fixed integer k? We show that G _n ^(p) and SL _n (F) are generated by their elements of order p and that the bound k exists and is equal to 4. We show that every universal p-Coxeter group has faithful two-dimensional representations over many fields F (including ? and ?). For a universal p-Coxeter group of rank ≥ 2 for p ≥ 3 or of rank ≥ 3 for p = 2 there is no bound k.     

Extra edge connectivity and isoperimetric edge connectivity

An edge set S of a connected graph G is a k-extra edge cut, if G-S is no longer connected, and each component of G-S has at least k vertices. The cardinality of a minimum k-extra edge cut, denoted by λ_k(G), is the k-extra edge connectivity of G. The kth isoperimetric edge connectivity γ_k(G) is defined as , where ω(U) is the number of edges with one end in U and the other end in . Write β_k(G)=min{ω(U):U⊂V(G),|U|=k}. A graph G with is said to be γ_k-optimal.In this paper, we first prove that λ_k(G)=γ_k(G) if G is a regular graph with girth g?k/2. Then, we show that except for K_3,3 and K₄, a 3-regular vertex/edge transitive graph is γ_k-optimal if and only if its girth is at least k+2. Finally, we prove that a connected d-regular edge-transitive graph with d?6e_k(G)/k is γ_k-optimal, where e_k(G) is the maximum number of edges in a subgraph of G with order k.     

Upper bounds on the signed (k, k)-domatic number

Let G be a graph with vertex set V(G), and let f : V(G) → {?1, 1} be a two-valued function. If k ≥ 1 is an integer and ${\sum_{x\in N[v]} f(x) \ge k}$ for each ${v \in V(G)}$ , where N[v] is the closed neighborhood of v, then f is a signed k-dominating function on G. A set {f ₁,f ₂, . . . ,f _d } of distinct signed k-dominating functions on G with the property that ${\sum_{i=1}^d f_i(x) \le k}$ for each ${x \in V(G)}$ , is called a signed (k, k)-dominating family (of functions) on G. The maximum number of functions in a signed (k, k)-dominating family on G is the signed (k, k)-domatic number of G. In this article we mainly present upper bounds on the signed (k, k)-domatic number, in particular for regular graphs.     

Conditional connectivity of bubble sort graphs

A subset F ? V (G) is called an R ^k -vertex-cut of a graph G if G ? F is disconnected and each vertex of G ? F has at least k neighbors in G ? F. The R ^k -vertex-connectivity of G, denoted by κ ^k (G), is the cardinality of a minimum R ^k -vertex-cut of G. Let B _n be the bubble sort graph of dimension n. It is known that κ _k (B _n ) = 2 ^k (n ? k ? 1) for n ≥ 2k and k = 1, 2. In this paper, we prove it for k = 3 and conjecture that it is true for all k ∈ N. We also prove that the connectivity cannot be more than conjectured.     

Cubicity and Bandwidth

A unit cube in ${\mathbb{R}^k}$ (or a k-cube in short) is defined as the Cartesian product R ₁ × R ₂?× ... × R _k where R _i (for 1??? i??? k) is a closed interval of the form [a _i , a _i + 1] on the real line. A k-cube representation of a graph G is a mapping of the vertices of G to k-cubes such that two vertices in G are adjacent if and only if their corresponding k-cubes have a non-empty intersection. The cubicity of G is the minimum k such that G has a k-cube representation. From a geometric embedding point of view, a k-cube representation of G?=?(V, E) yields an embedding ${f: V(G) \rightarrow \mathbb{R}^k}$ such that for any two vertices u and v, ||f(u) ? f(v)||_?? ?? 1 if and only if ${(u, v) \in E(G)}$ . We first present a randomized algorithm that constructs the cube representation of any graph on n vertices with maximum degree ?? in O(?? ln n) dimensions. This algorithm is then derandomized to obtain a polynomial time deterministic algorithm that also produces the cube representation of the input graph in the same number of dimensions. The bandwidth ordering of the graph is studied next and it is shown that our algorithm can be improved to produce a cube representation of the input graph G in O(?? ln b) dimensions, where b is the bandwidth of G, given a bandwidth ordering of G. Note that b ?? n and b is much smaller than n for many well-known graph classes. Another upper bound of b + 1 on the cubicity of any graph with bandwidth b is also shown. Together, these results imply that for any graph G with maximum degree ?? and bandwidth b, the cubicity is O(min{b, ?? ln b}). The upper bound of b?+ 1 is used to derive upper bounds for the cubicity of circular-arc graphs, cocomparability graphs and AT-free graphs in terms of the maximum degree ??.