On DP-coloring of graphs and multigraphs

While solving a question on the list coloring of planar graphs, Dvo?ák and Postle introduced the new notion of DP-coloring (they called it correspondence coloring). A DP-coloring of a graph G reduces the problem of finding a coloring of G from a given list L to the problem of finding a “large” independent set in the auxiliary graph H(G,L) with vertex set {(v, c): v ∈ V (G) and c ∈ L(v)}. It is similar to the old reduction by Plesnevi? and Vizing of the k-coloring problem to the problem of finding an independent set of size |V(G)| in the Cartesian product G□K _k, but DP-coloring seems more promising and useful than the Plesnevi?–Vizing reduction. Some properties of the DP-chromatic number χ _DP (G) resemble the properties of the list chromatic number χ _l (G) but some differ quite a lot. It is always the case that χ _DP (G) ≥ χ _l (G). The goal of this note is to introduce DP-colorings for multigraphs and to prove for them an analog of the result of Borodin and Erd?s–Rubin–Taylor characterizing the multigraphs that do not admit DP-colorings from some DP-degree-lists. This characterization yields an analog of Gallai’s Theorem on the minimum number of edges in n-vertex graphs critical with respect to DP-coloring.     

Chromatic number and subtrees of graphs

Let G and H be two graphs. We say that G induces H if G has an induced subgraph isomorphic to H: A. Gyárfás and D. Sumner, independently, conjectured that, for every tree T. there exists a function f _T ; called binding function, depending only on T with the property that every graph G with chromatic number f _T (ω(G)) induces T. A. Gyárfás, E. Szemerédi and Z. Tuza confirmed the conjecture for all trees of radius two on triangle-free graphs, and H. Kierstead and S. Penrice generalized the approach and the conclusion of A. Gyárfás et al. onto general graphs. A. Scott proved an interesting topological version of this conjecture asserting that for every integer k and every tree T of radius r, every graph G with ω(G) ? k and sufficient large chromatic number induces a subdivision of T of which each edge is subdivided at most O(14 ^r-1(r - 1)!) times. We extend the approach of A. Gyárfás and present a binding function for trees obtained by identifying one end of a path and the center of a star. We also improve A. Scott's upper bound by modifying his subtree structure and partition technique, and show that for every integer k and every tree T of radius r, every graph with ω(G) ? k and sufficient large chromatic number induces a subdivision of T of which each edge is subdivided at most O(6 ^r?2) times.     

A new characterization of symmetric group by NSE

Let G be a group and ω(G) be the set of element orders of G. Let k ∈ ω(G) and m _k (G) be the number of elements of order k in G. Let nse(G) = {m _k (G): k ∈ ω(G)}. Assume r is a prime number and let G be a group such that nse(G) = nse(S _r ), where S _r is the symmetric group of degree r. In this paper we prove that G ? S _r , if r divides the order of G and r ² does not divide it. To get the conclusion we make use of some well-known results on the prime graphs of finite simple groups and their components.     

Edge-colouring of graphs and hereditary graph properties

Edge-colourings of graphs have been studied for decades. We study edge-colourings with respect to hereditary graph properties. For a graph G, a hereditary graph property P and l ? 1 we define \(X{'_{P,l}}\)(G) to be the minimum number of colours needed to properly colour the edges of G, such that any subgraph of G induced by edges coloured by (at most) l colours is in P. We present a necessary and sufficient condition for the existence of \(X{'_{P,l}}\)(G). We focus on edge-colourings of graphs with respect to the hereditary properties O_k and S_k, where O_k contains all graphs whose components have order at most k+1, and S_k contains all graphs of maximum degree at most k. We determine the value of \(X{'_{{S_k},l}}(G)\) for any graph G, k ? 1, l ? 1, and we present a number of results on \(X{'_{{O_k},l}}(G)\).     

Distance domination of generalized de Bruijn and Kautz digraphs

Let G = (V,A) be a digraph and k ≥ 1 an integer. For u, v ∈ V, we say that the vertex u distance k-dominate v if the distance from u to v at most k. A set D of vertices in G is a distance k-dominating set if each vertex of V D is distance k-dominated by some vertex of D. The distance k-domination number of G, denoted by γ _k (G), is the minimum cardinality of a distance k-dominating set of G. Generalized de Bruijn digraphs G _B (n, d) and generalized Kautz digraphs G _K (n, d) are good candidates for interconnection networks. Denote Δ _k := (∑ _j=0 ^k d ^j )^?1. F. Tian and J. Xu showed that ?nΔ _k ? γ _k (G _B (n, d)) ≤?n/d ^k? and ?nΔ _k ? ≤ γ _k (G _K (n, d)) ≤ ?n/d ^k ?. In this paper, we prove that every generalized de Bruijn digraph G _B (n, d) has the distance k-domination number ?nΔ _k ? or ?nΔ _k ?+1, and the distance k-domination number of every generalized Kautz digraph G _K (n, d) bounded above by ?n/(d ^k?1+d ^k )?. Additionally, we present various sufficient conditions for γ _k (G _B (n, d)) = ?nΔ _k ? and γ _k (G _K (n, d)) = ?nΔ _k ?.     

Token Graphs

For a graph G and integer k ≥ 1, we define the token graph F _k (G) to be the graph with vertex set all k-subsets of V(G), where two vertices are adjacent in F _k (G) whenever their symmetric difference is a pair of adjacent vertices in G. Thus vertices of F _k (G) correspond to configurations of k indistinguishable tokens placed at distinct vertices of G, where two configurations are adjacent whenever one configuration can be reached from the other by moving one token along an edge from its current position to an unoccupied vertex. This paper introduces token graphs and studies some of their properties including: connectivity, diameter, cliques, chromatic number, Hamiltonian paths, and Cartesian products of token graphs.     

Length-type parameters of finite groups with almost unipotent automorphisms

Let α be an automorphism of a finite group G. For a positive integer n, let E _G,n (α) be the subgroup generated by all commutators [...[[x,α],α],…,α] in the semidirect product G 〈α〉 over x ∈ G, where α is repeated n times. By Baer’s theorem, if E _G,n (α)=1, then the commutator subgroup [G,α] is nilpotent. We generalize this theorem in terms of certain length parameters of E _G,n (α). For soluble G we prove that if, for some n, the Fitting height of E _G,n (α) is equal to k, then the Fitting height of [G,α] is at most k + 1. For nonsoluble G the results are in terms of the nonsoluble length and generalized Fitting height. The generalized Fitting height h*(H) of a finite group H is the least number h such that F _h* (H) = H, where F ₀* (H) = 1, and F _i+1* (H) is the inverse image of the generalized Fitting subgroup F*(H/F _i *(H)). Let m be the number of prime factors of the order |α| counting multiplicities. It is proved that if, for some n, the generalized Fitting height E _G,n (α) of is equal to k, then the generalized Fitting height of [G,α] is bounded in terms of k and m. The nonsoluble length λ(H) of a finite group H is defined as the minimum number of nonsoluble factors in a normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. It is proved that if λE _G,n (α)= k, then the nonsoluble length of [G,α] is bounded in terms of k and m. We also state conjectures of stronger results independent of m and show that these conjectures reduce to a certain question about automorphisms of direct products of finite simple groups.     

Engel-type subgroups and length parameters of finite groups

Let g be an element of a finite group G. For a positive integer n, let E _n (g) be the subgroup generated by all commutators [...[[x, g], g],..., g] over x ∈ G, where g is repeated n times. By Baer’s theorem, if E _n (g) = 1, then g belongs to the Fitting subgroup F(G). We generalize this theorem in terms of certain length parameters of E _n (g). For soluble G we prove that if, for some n, the Fitting height of E _n (g) is equal to k, then g belongs to the (k+1)th Fitting subgroup F_k+1(G). For nonsoluble G the results are in terms of nonsoluble length and generalized Fitting height. The generalized Fitting height h*(H) of a finite group H is the least number h such that F_h* (H) = H, where F₀* (H) = 1, and F_i+1(H)* is the inverse image of the generalized Fitting subgroup F*(H/F*_i (H)). Let m be the number of prime factors of |g| counting multiplicities. It is proved that if, for some n, the generalized Fitting height of E _n (g) is equal to k, then g belongs to F*_f(k,m)(G), where f(k, m) depends only on k and m. The nonsoluble length λ(H) of a finite group H is defined as the minimum number of nonsoluble factors in a normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. It is proved that if λ(E _n (g)) = k, then g belongs to a normal subgroup whose nonsoluble length is bounded in terms of k and m. We also state conjectures of stronger results independent of m and show that these conjectures reduce to a certain question about automorphisms of direct products of finite simple groups.     

On a problem of impulse control under interference

This paper contains several results about the structure of the congruence kernel C^(S)(G) of an absolutely almost simple simply connected algebraic group G over a global field K with respect to a set of places S of K. In particular, we show that C^(S)(G)) is always trivial if S contains a generalized arithmetic progression. We also give a criterion for the centrality of C^(S)(G) in the general situation in terms of the existence of commuting lifts of the groups G(K_v) for v ? S in the S-arithmetic completion ?^(S). This result enables one to give simple proofs of the centrality in a number of cases. Finally, we show that if K is a number field and G is K-isotropic, then C^(S)(G) as a normal subgroup of ?^(S) is almost generated by a single element.     

Minimum Difference Representations of Graphs

Define a k-minimum-difference-representation (k-MDR) of a graph G to be a family of sets \({\{S(v): v\in V(G)\}}\) such that u and v are adjacent in G if and only if min{|S(u)?S(v)|, |S(v)?S(u)|} ≥ k. Define ρ _min(G) to be the smallest k for which G has a k-MDR. In this note, we show that {ρ _min(G)} is unbounded. In particular, we prove that for every k there is an n ₀ such that for n > n ₀ ‘almost all’ graphs of order n satisfy ρ _min(G) > k. As our main tool, we prove a Ramsey-type result on traces of hypergraphs.     

New characterization of <Emphasis Type="Italic">S</Emphasis><Subscript>4</Subscript>(<Emphasis Type="Italic">q</Emphasis>) by its noncommuting graph

Let G be a nonabelian group, and associate the noncommuting graph ?(G) with G as follows: the vertex set of ?(G) is G\Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity. Let S ₄(q) be the projective symplectic simple group, where q is a prime power. We prove that if G is a group with ?(G) ? ?(S ₄(q)) then G ? S ₄(q).     

The Upper Connected Vertex Detour Monophonic Number of a Graph

For any vertex x in a connected graph G of order n ≥ 2, a set S _x ? V (G) is an x-detour monophonic set of G if each vertex v ∈ V (G) lies on an x-y detour monophonic path for some element y in S _x . The minimum cardinality of an x-detour monophonic set of G is the x-detour monophonic number of G, denoted by dm _x (G). A connected x-detour monophonic set of G is an x-detour monophonic set S _x such that the subgraph induced by S _x is connected. The minimum cardinality of a connected x-detour monophonic set of G is the connected x-detour monophonic number of G, denoted by cdm _x (G). A connected x-detour monophonic set S _x of G is called a minimal connected x-detour monophonic set if no proper subset of S _x is a connected x-detour monophonic set. The upper connected x-detour monophonic number of G, denoted by cdm⁺ _x (G), is defined to be the maximum cardinality of a minimal connected x-detour monophonic set of G. We determine bounds and exact values of these parameters for some special classes of graphs. We also prove that for positive integers r,d and k with 2 ≤ r ≤ d and k ≥ 2, there exists a connected graph G with monophonic radius r, monophonic diameter d and upper connected x-detour monophonic number k for some vertex x in G. Also, it is shown that for positive integers j,k,l and n with 2 ≤ j ≤ k ≤ l ≤ n - 3, there exists a connected graph G of order n with dm _x (G) = j,dm⁺ _x (G) = k and cdm⁺ _x (G) = l for some vertex x in G.     

Representations of the general linear group over symmetry classes of polynomials

Let V be the complex vector space of homogeneous linear polynomials in the variables x₁,..., x _m . Suppose G is a subgroup of S _m , and χ is an irreducible character of G. Let H _d (G, χ) be the symmetry class of polynomials of degree d with respect to G and χ.

For any linear operator T acting on V, there is a (unique) induced operator K _χ (T) ∈ End(H _d (G, χ)) acting on symmetrized decomposable polynomials by

$${K_\chi }\left( T \right)\left( {{f_1} * {f_2} * \cdots * {f_d}} \right) = T{f_1} * T{f_2} * \cdots * T{f_d}.$$

In this paper, we show that the representation T ? K _χ (T) of the general linear group GL(V) is equivalent to the direct sum of χ(1) copies of a representation (not necessarily irreducible) T ? B _χ ^G (T).

     

Neighbor sum distinguishing total chromatic number of <Emphasis Type="Italic">K</Emphasis><Subscript>4</Subscript>-minor free graph

A k-total coloring of a graph G is a mapping ?: V (G) ? E(G) → {1; 2,..., k} such that no two adjacent or incident elements in V (G) ? E(G) receive the same color. Let f(v) denote the sum of the color on the vertex v and the colors on all edges incident with v: We say that ? is a k-neighbor sum distinguishing total coloring of G if f(u) 6 ≠ f(v) for each edge uv ∈ E(G): Denote χ _Σ ^″ (G) the smallest value k in such a coloring of G: Pil?niak and Wo?niak conjectured that for any simple graph with maximum degree Δ(G), χ _Σ ^″ ≤ Δ(G)+3. In this paper, by using the famous Combinatorial Nullstellensatz, we prove that for K ₄-minor free graph G with Δ(G) > 5; χ _Σ ^″ = Δ(G) + 1 if G contains no two adjacent Δ-vertices, otherwise, χ _Σ ^″ (G) = Δ(G) + 2.     

On <Emphasis Type="Italic">k</Emphasis>- critical 2<Emphasis Type="Italic">k</Emphasis>- connected graphs

A graph G is called an (n,k)-graph if κ(G-S)=n-|S| for any S ? V(G) with |S| ≤ k, where ?(G) denotes the connectivity of G. Mader conjectured that for k ≥ 3 the graph K_2k+2?(1-factor) is the unique (2k, k)-graph. Kriesell has settled two special cases for k = 3,4. We prove the conjecture for the general case k ≥ 5.     

A Note on Generalized Lagrangians of Non-uniform Hypergraphs

Set \(A\subset {\mathbb N}\) is less than \(B\subset {\mathbb N}\) in the colex ordering if m a x(A△B)∈B. In 1980’s, Frankl and Füredi conjectured that the r-uniform graph with m edges consisting of the first m sets of \({\mathbb N}^{(r)}\) in the colex ordering has the largest Lagrangian among all r-uniform graphs with m edges. A result of Motzkin and Straus implies that this conjecture is true for r=2. This conjecture seems to be challenging even for r=3. For a hypergraph H=(V,E), the set T(H)={|e|:e∈E} is called the edge type of H. In this paper, we study non-uniform hypergraphs and define L(H) a generalized Lagrangian of a non-uniform hypergraph H in which edges of different types have different weights. We study the following two questions: 1. Let H be a hypergraph with m edges and edge type T. Let C _m,T denote the hypergraph with edge type T and m edges formed by taking the first m sets with cardinality in T in the colex ordering. Does L(H)≤L(C _m,T ) hold? If T={r}, then this question is the question by Frankl and Füredi. 2. Given a hypergraph H, find a minimum subhypergraph G of H such that L(G) = L(H). A result of Motzkin and Straus gave a complete answer to both questions if H is a graph. In this paper, we give a complete answer to both questions for {1,2}-hypergraphs. Regarding the first question, we give a result for {1,r ₁,r ₂,…,r _l }-hypergraph. We also show the connection between the generalized Lagrangian of {1,r ₁,r ₂,? ,r _l }-hypergraphs and {r ₁,r ₂,? ,r _l }-hypergraphs concerning the second question.     

Hopf Modules and Representations of Finite Wreath Products

For a finite group G and nonnegative integer n ≥ 0, one may consider the associated tower \(G \wr S_{n} := S_{n} \ltimes G^{n}\) of wreath product groups. Zelevinsky associated to such a tower the structure of a positive self-adjoint Hopf algebra (PSH-algebra) R(G) on the direct sum over integers n ≥ 0 of the Grothendieck groups K ₀(R e p?G?S _n ). In this paper, we study the interaction via induction and restriction of the PSH-algebras R(G) and R(H) associated to finite groups H ? G. A class of Hopf modules over PSH-algebras with a compatibility between the comultiplication and multiplication involving the Hopf k ^{t h} -power map arise naturally and are studied independently. We also give an explicit formula for the natural PSH-algebra morphisms R(H) → R(G) and R(G) → R(H) arising from induction and restriction. In an appendix, we consider a family of subgroups of wreath product groups analogous to the subgroups G(m, p, n) of the wreath product cyclotomic complex reflection groups G(m, 1, n).     

A variation of a conjecture due to Erdös and Sós

Erdoes and Soes conjectured in 1963 that every graph G on n vertices with edge number e（G） 〉 1/2（k - 1）n contains every tree T with k edges as a subgraph. In this paper, we consider a variation of the above conjecture, that is, for n 〉 9/ 2k^2 ＋ 37/2＋ 14 and every graph G on n vertices with e（G） 〉 1/2 （k- 1）n, we prove that there exists a graph G＇ on n vertices having the same degree sequence as G and containing every tree T with k edges as a subgraph.     

Bivariate Binomial Moments and Bonferroni-Type Inequalities

We obtain bivariate forms of Gumbel’s, Fréchet’s and Chung’s linear inequalities for P(S ≥ u, T ≥ v) in terms of the bivariate binomial moments {S _{i, j} }, 1 ≤ i ≤ k,1 ≤ j ≤ l of the joint distribution of (S, T). At u = v = 1, the Gumbel and Fréchet bounds improve monotonically with non-decreasing (k, l). The method of proof uses combinatorial identities, and reveals a multiplicative structure before taking expectation over sample points.     

A construction for modified generalized Hadamard matrices using QGH matrices

Let G be a group of order mu and U a normal subgroup of G of order u. Let G/U = {U ₁,U ₂, . . . ,U _m } be the set of cosets of U in G. We say a matrix H = [h _ij ] of order k with entries from G is a quasi-generalized Hadamard matrix with respect to the cosets G/U if \({\sum_{1\le t \le k} h_{it}h_{jt}^{-1} = \lambda_{ij1}U_1+\cdots+\lambda_{ijm}U_m (\exists\lambda_{ij1},\ldots, \exists \lambda_{ijm} \in \mathbb{Z})}\) for any i ≠ j. On the other hand, in our previous article we defined a modified generalized Hadamard matrix GH(s, u, λ) over a group G, from which a TD _λ (uλ, u) admitting G as a semiregular automorphism group is obtained. In this article, we present a method for combining quasi-generalized Hadamard matrices and semiregular relative difference sets to produce modified generalized Hadamard matrices.