How to play the Majority game with a liar

The Majority game is played by a questioner () and an answerer (). holds n elements, each of which can be labeled as 0 or 1. is trying to identify some element holds as having the Majority label or, in the case of a tie, claim there is none. To do this asks questions comparing whether two elements have the same or different label. ’s goal is to ask as few questions as possible while ’s goal is to delay as much as possible. Let q^∗ denote the minimal number of questions needed for to identify a Majority element regardless of ’s answers.In this paper we investigate upper and lower bounds for q^∗ in a variation of the Majority game, where is allowed to lie up to t times. We consider two versions of the game, the adaptive (where questions are asked sequentially) and the oblivious (where questions are asked in one batch).     

On wreathed lexicographic products of graphs

This paper proves a necessary and sufficient condition for the endomorphism monoid of a lexicographic product G[H] of graphs G,H to be the wreath product of the monoids and . The paper also gives respective necessary and sufficient conditions for specialized cases such as for unretractive or triangle-free graphs G.     

Posets of annular non-crossing partitions of types B and D

We study the set of annular non-crossing permutations of type B, and we introduce a corresponding set of annular non-crossing partitions of type B, where p and q are two positive integers. We prove that the natural bijection between and is a poset isomorphism, where the partial order on is induced from the hyperoctahedral group B_p+q, while is partially ordered by reverse refinement. In the case when q=1, we prove that is a lattice with respect to reverse refinement order.We point out that an analogous development can be pursued in type D, where one gets a canonical isomorphism between and . For q=1, the poset coincides with a poset “NC^(D)(p+1)” constructed in a paper by Athanasiadis and Reiner [C.A. Athanasiadis, V. Reiner, Noncrossing partitions for the group D_n, SIAM Journal of Discrete Mathematics 18 (2004) 397-417], and is a lattice by the results of that paper.     

On the number of graphs with a given endomorphism monoid

For a given finite monoid , let be the number of graphs on n vertices with endomorphism monoid isomorphic to . For any nontrivial monoid we prove that where and are constants depending only on with .For every k there exists a monoid of size k with , on the other hand if a group of unity of has a size k>2 then .     

Generalization of transitive fraternal augmentations for directed graphs and its applications

Let be a directed graph. A transitive fraternal augmentation of is a directed graph with the same vertex set, including all the arcs of and such that for any vertices x,y,z,

1.

if and then or (fraternity);

2.

if and then (transitivity).

In this paper, we explore some generalization of the transitive fraternal augmentations for directed graphs and its applications. In particular, we show that the 2-coloring number col₂(G)≤O(∇₁(G)∇₀(G)²), where ∇_k(G) (k≥0) denotes the greatest reduced average density with depth k of a graph G; we give a constructive proof that ∇_k(G) bounds the distance (k+1)-coloring number col_k+1(G) with a function f(∇_k(G)). On the other hand, ∇_k(G)≤(col_2k+1(G))^2k+1. We also show that an inductive generalization of transitive fraternal augmentations can be used to study nonrepetitive colorings of graphs.     

Constructions of optimal quaternary constant weight codes via group divisible designs

Generalized Steiner systems were first introduced by Etzion and used to construct optimal constant weight codes over an alphabet of size g+1 with minimum Hamming distance 2k−3, in which each codeword has length v and weight k. As to the existence of a , a lot of work has been done for k=3, while not so much is known for k=4. The notion k-^∗GDD was first introduced by Chen et al. and used to construct . The necessary condition for the existence of a is v≥14. In this paper, it is proved that there exists a for any prime power and v≥19. By using this result, the known results on the existence of optimal quaternary constant weight codes are then extended.     

On cyclic semi-regular subgroups of certain 2-transitive permutation groups

We determine the cyclic semi-regular subgroups of the 2-transitive permutation groups and with n a suitable power of a prime number p.     

A note on an embedding problem in transitive tournaments

Let TT_n be a transitive tournament on n vertices. It is known Görlich, Pil?niak, Wo?niak, (2006) [3] that for any acyclic oriented graph of order n and size not greater than , two graphs isomorphic to are arc-disjoint subgraphs of TT_n. In this paper, we consider the problem of embedding of acyclic oriented graphs into their complements in transitive tournaments. We show that any acyclic oriented graph of size at most is embeddable into all its complements in TT_n. Moreover, this bound is generally the best possible.     

Some relations between rank, chromatic number and energy of graphs

The energy of a graph G, denoted by E(G), is defined as the sum of the absolute values of all eigenvalues of G. Let G be a graph of order n and be the rank of the adjacency matrix of G. In this paper we characterize all graphs with . Among other results we show that apart from a few families of graphs, , where n is the number of vertices of G, and χ(G) are the complement and the chromatic number of G, respectively. Moreover some new lower bounds for E(G) in terms of are given.     

Large monochromatic components in colorings of complete 3-uniform hypergraphs

Let f(n,r) be the largest integer m with the following property: if the edges of the complete 3-uniform hypergraph are colored with r colors then there is a monochromatic component with at least m vertices. Here we show that and . Both results are sharp under suitable divisibility conditions (namely if n is divisible by 7, or by 6 respectively).     

Sequences of integers with missing quotients

Let S be any set of natural numbers, and A be a given set of rational numbers. We say that S is an A-quotient-free set if x,y∈S implies y/x∉A. Let and , where the supremum is taken over all A-quotient-free sets S, and are the upper and lower asymptotic densities of S respectively. Let ρ(A)=sup_Sδ(S), where the supremum is taken over all A-quotient-free sets S such that δ(S) exists. In this paper we study the properties of , and ρ(A).     

More on almost self-complementary graphs

A graph X is called almost self-complementary if it is isomorphic to one of its almost complements , where denotes the complement of X and I a perfect matching (1-factor) in . If I is a perfect matching in and is an isomorphism, then the graph X is said to be fairly almost self-complementary if φ preserves I setwise, and unfairly almost self-complementary if it does not.In this paper we construct connected graphs of all possible orders that are fairly and unfairly almost self-complementary, fairly but not unfairly almost self-complementary, and unfairly but not fairly almost self-complementary, respectively, as well as regular graphs of all possible orders that are fairly and unfairly almost self-complementary.Two perfect matchings I and J in are said to be X-non-isomorphic if no isomorphism from X+I to X+J induces an automorphism of X. We give a constructive proof to show that there exists a graph X that is almost self-complementary with respect to two X-non-isomorphic perfect matchings for every even order greater than or equal to four.     

All countable monoids embed into the monoid of the infinite random graph

We prove that the full transformation monoid on a countably infinite set is isomorphic to a submonoid of , the endomorphism monoid of the infinite random graph R. Consequently, embeds each countable monoid, satisfies no nontrivial monoid identity, and has an undecidable universal theory.     

On the relation between cluster and classical tilting

Let be a triangulated category with a cluster tilting subcategory U. The quotient category is abelian; suppose that it has finite global dimension.We show that projection from to sends cluster tilting subcategories of to support tilting subcategories of , and that, in turn, support tilting subcategories of can be lifted uniquely to weak cluster tilting subcategories of .     

Higher Kurtz randomness

A real x is -Kurtz random (-Kurtz random) if it is in no closed null set ( set). We show that there is a cone of -Kurtz random hyperdegrees. We characterize lowness for -Kurtz randomness as being -dominated and -semi-traceable.     

On the Wiener integral with respect to a sub-fractional Brownian motion on an interval

The domain of the Wiener integral with respect to a sub-fractional Brownian motion , , k≠0, is characterized. The set is a Hilbert space which contains the class of elementary functions as a dense subset. If , any element of is a function and if , the domain is a space of distributions.     

Mixed Artin-Tate motives over number rings

This paper studies Artin-Tate motives over bases , for a number field F. As a subcategory of motives over S, the triangulated category of Artin-Tate motives is generated by motives , where ? is any finite map. After establishing the stability of these subcategories under pullback and pushforward along open and closed immersions, a motivic t-structure is constructed. Exactness properties of these functors familiar from perverse sheaves are shown to hold in this context. The cohomological dimension of mixed Artin-Tate motives () is two, and there is an equivalence .     

Game colouring of the square of graphs

This paper studies the game chromatic number and game colouring number of the square of graphs. In particular, we prove that if G is a forest of maximum degree Δ≥9, then , and there are forests G with . It is also proved that for an outerplanar graph G of maximum degree Δ, , and for a planar graph G of maximum degree Δ, .     

Total domination and total domination subdivision number of a graph and its complement

A set S of vertices of a graph G=(V,E) with no isolated vertex is a total dominating set if every vertex of V(G) is adjacent to some vertex in S. The total domination numberγ_t(G) is the minimum cardinality of a total dominating set of G. The total domination subdivision numbersd_γt(G) is the minimum number of edges that must be subdivided in order to increase the total domination number. We consider graphs of order n?4, minimum degree δ and maximum degree Δ. We prove that if each component of G and has order at least 3 and , then and if each component of G and has order at least 2 and at least one component of G and has order at least 3, then . We also give a result on stronger than a conjecture by Harary and Haynes.