Relaxed very asymmetric coloring games

This paper investigates a competitive version of the coloring game on a finite graph G. An asymmetric variant of the (r,d)-relaxed coloring game is called the (r,d)-relaxed (a,b)-coloring game. In this game, two players, Alice and Bob, take turns coloring the vertices of a graph G, using colors from a set X, with |X|=r. On each turn Alice colors a vertices and Bob colors b vertices. A color α∈X is legal for an uncolored vertex u if by coloring u with color α, the subgraph induced by all the vertices colored with α has maximum degree at most d. Each player is required to color an uncolored vertex legally on each move. The game ends when there are no remaining uncolored vertices. Alice wins the game if all vertices of the graph are legally colored, Bob wins if at a certain stage there exists an uncolored vertex without a legal color. The d-relaxed (a,b)-game chromatic number, denoted by , of G is the least r for which Alice has a winning strategy in the (r,d)-relaxed (a,b)-coloring game.The (r,d)-relaxed (1,1)-coloring game has been well studied and there are many interesting results. For the (r,d)-relaxed (a,1)-coloring game, this paper proves that if a graph G has an orientation with maximum outdegree k and a≥k, then for all d≥k²+2k; If a≥k³, then (a,1)- for all d≥2k+1.     

Circular mixed hypergraphs II:The upper chromatic number

A mixed hypergraph is a triple H=(X,C,D), where X is the vertex set and each of C, D is a family of subsets of X, the C-edges and D-edges, respectively. A proper k-coloring of H is a mapping c:X→[k] such that each C-edge has two vertices with a common color and each D-edge has two vertices with distinct colors. A mixed hypergraph H is called circular if there exists a host cycle on the vertex set X such that every edge (C- or D-) induces a connected subgraph of this cycle.We suggest a general procedure for coloring circular mixed hypergraphs and prove that if H is a reduced colorable circular mixed hypergraph with n vertices, upper chromatic number and sieve number s, then

     

Domination in a digraph and in its reverse

Let D be a digraph. By γ(D) we denote the domintaion number of D and by D⁻ we denote a digraph obtained by reversing all the arcs of D. In this paper we prove that for every δ≥3 and k≥1 there exists a simple strongly connected δ-regular digraph D_δ,k such that . Analogous theorem is obtained for total domination number provided that δ≥4.     

Weak acyclic coloring and asymmetric coloring games

We introduce the notion of weak acyclic coloring of a graph. This is a relaxation of the usual notion of acyclic coloring which is often sufficient for applications. We then use this concept to analyze the (a,b)-coloring game. This game is played on a finite graph G, using a set of colors X, by two players Alice and Bob with Alice playing first. On each turn Alice (Bob) chooses a (b) uncolored vertices and properly colors them with colors from X. Alice wins if the players eventually create a proper coloring of G; otherwise Bob wins when one of the players has no legal move. The (a,b)-game chromatic number of G, denoted (a,b)-χ_g(G), is the least integer t such that Alice has a winning strategy when the game is played on G using t colors. We show that if the weak acyclic chromatic number of G is at most k then (2,1)-.     

Randomized strategies for the plurality problem

We consider a game played by two players, Paul and Carol. At the beginning of the game, Carol fixes a coloring of n balls. At each turn, Paul chooses a pair of the balls and asks Carol whether the balls have the same color. Carol truthfully answers his question. Paul’s goal is to determine the most frequent (plurality) color in the coloring by asking as few questions as possible. The game is studied in the probabilistic setting when Paul is allowed to choose his next question randomly.We give asymptotically tight bounds both for the case of two colors and many colors. For the balls colored by k colors, we prove a lower bound Ω(kn) on the expected number of questions; this is asymptotically optimal. For the balls colored by two colors, we provide a strategy for Paul to determine the plurality color with the expected number of questions; this almost matches the lower bound .     

The partition bargaining problem

Consider the problem of partitioning n items among d players where the utility of each player for bundles of items is additive; so, player r has utility for item i and the utility of that player for a bundle of items is the sum of the 's over the items i in his/her bundle. Each partition S of the items is then associated with a d-dimensional utility vector V^S whose coordinates are the utilities that the players assign to the bundles they get under S. Also, lotteries over partitions are associated with the corresponding expected utility vectors. We model the problem as a Nash bargaining game over the set of lotteries over partitions and provide methods for computing the corresponding Nash solution, to prescribed accuracy, with effort that is polynomial in n. In particular, we show that points in the pareto-optimal set of the corresponding bargaining set correspond to lotteries over partitions under which each item, with the possible exception of at most d(d-1)/2 items, is assigned in the same way.     

On hamiltonian colorings for some graphs

For a connected graph G and any two vertices u and v in G, let D(u,v) denote the length of a longest u-v path in G. A hamiltonian coloring of a connected graph G of order n is an assignment c of colors (positive integers) to the vertices of G such that |c(u)−c(v)|+D(u,v)≥n−1 for every two distinct vertices u and v in G. The value of a hamiltonian coloring c is the maximum color assigned to a vertex of G. The hamiltonian chromatic number of G is taken over all hamiltonian colorings c of G. In this paper we discuss the hamiltonian chromatic number of graphs G with . As examples, we determine the hamiltonian chromatic number for a class of caterpillars, and double stars.     

Any Diophantine quintuple contains a regular Diophantine quadruple

A set {a₁,…,am} of m distinct positive integers is called a Diophantine m-tuple if aiaj+1 is a perfect square for all i, j with 1?i<j?m. It is conjectured that if {a,b,c,d} is a Diophantine quadruple with a<b<c<d, then d=d₊, where d₊=a+b+c+2abc+2rst and , , . In this paper, we show that if {a,b,c,d,e} is a Diophantine quintuple with a<b<c<d<e, then d=d₊.     

Inducing regulation of any digraphs

For a given structure D (digraph, multidigraph, or pseudodigraph) and an integer r large enough, a smallest inducing r-regularization of D is constructed. This regularization is an r-regular superstructure of the smallest possible order with bounded arc multiplicity, and containing D as an induced substructure. The sharp upper bound on the number, ρ, of necessary new vertices among such superstructures for n-vertex general digraphs D is determined, ρ being called the inducing regulation number of D. For being the maximum among semi-degrees in D, simple n-vertex digraphs D with largest possible ρ are characterized if either or (where the case is not a trivial subcase of ).     

Distance-two labelings of digraphs

For positive integers j?k, an L(j,k)-labeling of a digraph D is a function f from V(D) into the set of nonnegative integers such that |f(x)-f(y)|?j if x is adjacent to y in D and |f(x)-f(y)|?k if x is of distance two to y in D. Elements of the image of f are called labels. The L(j,k)-labeling problem is to determine the -number of a digraph D, which is the minimum of the maximum label used in an L(j,k)-labeling of D. This paper studies -numbers of digraphs. In particular, we determine -numbers of digraphs whose longest dipath is of length at most 2, and -numbers of ditrees having dipaths of length 4. We also give bounds for -numbers of bipartite digraphs whose longest dipath is of length 3. Finally, we present a linear-time algorithm for determining -numbers of ditrees whose longest dipath is of length 3.     

The 6-relaxed game chromatic number of outerplanar graphs

Suppose G is a graph and k,d are integers. The (k,d)-relaxed colouring game on G is a game played by two players, Alice and Bob, who take turns colouring the vertices of G with legal colours from a set X of k colours. Here a colour i is legal for an uncoloured vertex x if after colouring x with colour i, the subgraph induced by vertices of colour i has maximum degree at most d. Alice’s goal is to have all the vertices coloured, and Bob’s goal is the opposite: to have an uncoloured vertex without a legal colour. The d-relaxed game chromatic number of G, denoted by , is the least number k so that when playing the (k,d)-relaxed colouring game on G, Alice has a winning strategy. This paper proves that if G is an outerplanar graph, then for d≥6.     

Singular values, norms, and commutators

Let and X_i, i=1,…,n, be bounded linear operators on a separable Hilbert space such that X_i is compact for i=1,…,n. It is shown that the singular values of are dominated by those of , where ‖·‖ is the usual operator norm. Among other applications of this inequality, we prove that if A and B are self-adjoint operators such that a₁?A?a₂ and b₁?B?b₂ for some real numbers and b₂, and if X is compact, then the singular values of the generalized commutator AX-XB are dominated by those of max(b₂-a₁,a₂-b₁)(X⊕X). This inequality proves a recent conjecture concerning the singular values of commutators. Several inequalities for norms of commutators are also given.     

Equitable colorings of Kronecker products of graphs

For a positive integer k, a graph G is equitably k-colorable if there is a mapping f:V(G)→{1,2,…,k} such that f(x)≠f(y) whenever xy∈E(G) and ||f⁻¹(i)|−|f⁻¹(j)||≤1 for 1≤i<j≤k. The equitable chromatic number of a graph G, denoted by χ₌(G), is the minimum k such that G is equitably k-colorable. The equitable chromatic threshold of a graph G, denoted by , is the minimum t such that G is equitably k-colorable for k≥t. The current paper studies equitable chromatic numbers of Kronecker products of graphs. In particular, we give exact values or upper bounds on χ₌(G×H) and when G and H are complete graphs, bipartite graphs, paths or cycles.     

The upper connected geodetic number and forcing connected geodetic number of a graph

For a connected graph G of order p≥2, a set S⊆V(G) is a geodetic set of G if each vertex v∈V(G) lies on an x-y geodesic for some elements x and y in S. The minimum cardinality of a geodetic set of G is defined as the geodetic number of G, denoted by g(G). A geodetic set of cardinality g(G) is called a g-set of G. A connected geodetic set of G is a geodetic set S such that the subgraph G[S] induced by S is connected. The minimum cardinality of a connected geodetic set of G is the connected geodetic number of G and is denoted by g_c(G). A connected geodetic set of cardinality g_c(G) is called a g_c-set of G. A connected geodetic set S in a connected graph G is called a minimal connected geodetic set if no proper subset of S is a connected geodetic set of G. The upper connected geodetic number is the maximum cardinality of a minimal connected geodetic set of G. We determine bounds for and determine the same for some special classes of graphs. For positive integers r,d and n≥d+1 with r≤d≤2r, there exists a connected graph G with , and . Also, for any positive integers 2≤a<b≤c, there exists a connected graph G such that g(G)=a, g_c(G)=b and . A subset T of a g_c-set S is called a forcing subset for S if S is the unique g_c-set containing T. A forcing subset for S of minimum cardinality is a minimum forcing subset of S. The forcing connected geodetic number of S, denoted by f_c(S), is the cardinality of a minimum forcing subset of S. The forcing connected geodetic number of G, denoted by f_c(G), is f_c(G)=min{f_c(S)}, where the minimum is taken over all g_c-sets S in G. It is shown that for every pair a,b of integers with 0≤a≤b−4, there exists a connected graph G such that f_c(G)=a and g_c(G)=b.     

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).