Tsai-Lien Wong

Total weight choosability of graphs

A graph G = (V, E) is called (k, k′)-total weight choosable if the following holds: For any total list assignment L which assigns to each vertex x a set L(x) of k real numbers, and assigns to each edge e a set L(e) of k′ real numbers, there is a mapping f: V∪E→ℝ such that f(y)∈L(y) for any y∈V∪Eand for any two adjacent vertices x, x′, **image**. We conjecture that every graph is (2, 2)-total weight choosable and every graph without isolated edges is (1, 3)-total weight choosable. It follows from results in [7] that complete graphs, complete bipartite graphs, trees other than K2 are (1, 3)-total weight choosable. Also a graph G obtained from an arbitrary graph H by subdividing each edge with at least three vertices is (1, 3)-total weight choosable. This article proves that complete graphs, trees, generalized theta graphs are (2, 2)-total weight choosable. We also prove that for any graph H, a graph G obtained from H by subdividing each edge with at least two vertices is (2, 2)-total weight choosable as well as (1, 3)-total weight choosable.

Antimagic labelling of vertex weighted graphs

Suppose G is a graph, k is a non-negative integer. We say G is k-antimagic if there is an injection f: E→{1, 2, …, |E| + k} such that for any two distinct vertices u and v, . We say G is weighted-k-antimagic if for any vertex weight function w: V→ℕ, there is an injection f: E→{1, 2, …, |E| + k} such that for any two distinct vertices u and v, . A well-known conjecture asserts that every connected graph G≠K2 is 0-antimagic. On the other hand, there are connected graphs G≠K2 which are not weighted-1-antimagic. It is unknown whether every connected graph G≠K2 is weighted-2-antimagic. In this paper, we prove that if G has a universal vertex, then G is weighted-2-antimagic. If G has a prime number of vertices and has a Hamiltonian path, then G is weighted-1-antimagic. We also prove that every connected graph G≠K2 on n vertices is weighted- ⌊3n/2⌋-antimagic. Copyright

Every graph is (2,3)-choosable

A total weighting of a graph G is a mapping ϕ that assigns to each element z ∈ V (G)∪E(G) a weight ϕ(z). A total weighting ϕ is proper if for any two adjacent vertices u and v, ∑e∈E(u)ϕ(e)+ϕ(u)≠∑e∈E(v)ϕ(e)+ϕ(v). This paper proves that if each edge e is given a set L(e) of 3 permissible weights, and each vertex v is given a set L(v) of 2 permissible weights, then G has a proper total weighting ϕ with ϕ(z) ∈ L(z) for each element z ∈ V (G)∪E(G).

List Total Weighting of Graphs

A graph G=(V, E) is (k, k′)-total weight choosable if the following is true: For any (k, k′)-total list assignment L that assigns to each vertex v a set L(v) of k real numbers as permissible weights, and assigns to each edge e a set L(e) of k′ real numbers as permissible weights, there is a proper L-total weighting, i.e., a mapping f:V∪E→ℝ such that f(y)∈L(y) for each y∈V∪E, and for any two adjacent vertices u and v, ∑ e∈E(u) f(e)+f(u)≠∑ e∈E(v) f(e)+f(v). This Paper introduces a method, the max-min weighting method, for finding proper L-total weightings of graphs. Using this method, we prove that complete multipartite graphs of the form K n,m,1,1,...,1 are (2,2)-total weight choosable and complete bipartite graphs other than K 2 are (1,2)-total weight choosable.

Semiprime Algebras with Finiteness Conditions

Abstract Smyth proved that given an element x in a semiprime Banach algebra B, xB is finite-dimensional if and only if Bx is finite-dimensional. In a recent paper, Yood extended the result to arbitrary semiprime algebras. In this paper we study unital one-sided ideals in semiprime algebras from different viewpoints and prove several results on equalities of dimensions. Finally, we give a structure theorem of semiprime algebras whose zero subalgebras are finite-dimensional. All arguments concerning algebras can be easily modified to the case of rings.

Nonadditive Strong Commutativity Preserving Maps

The goal of this note is to characterize strong commutativity preserving maps on prime or semiprime rings without the additivity assumption on the maps considered in the literature.

A NOTE ON DERIVATIONS, ADDITIVE SUBGROUPS, AND LIE IDEALS OF PRIME RINGS

ABSTRACT Let be a prime ring of characteristic not or and a noncentral Lie ideal of . If is a nonzero derivation of , then the additive subgroup of generated by the subset contains a noncentral Lie ideal of .

ON CERTAIN SUBGROUPS OF PRIME RINGS WITH AUTOMORPHISMS

ABSTRACT Let be a noncommutative prime ring and an automorphism of , . We denote by the additive subgroup of generated by the subset and by the subring of generated by . In this paper we prove that if , then contains for some nonzero ideal of . Also, we prove the following theorem concerning : Suppose that . Then contains a nonzero ideal of if and only if either or .

Adaptable chromatic number of graph products

A colouring of the vertices of a graph (or hypergraph) G is adapted to a given colouring of the edges of G if no edge has the same colour as both (or all) its vertices. The adaptable chromatic number of G is the smallest integer k such that each edge-colouring of G by colours 1,2,...,k admits an adapted vertex-colouring of G by the same colours 1,2,...,k. (The adaptable chromatic number is just one more than a previously investigated notion of chromatic capacity.) The adaptable chromatic number of a graph G is smaller than or equal to the ordinary chromatic number of G. While the ordinary chromatic number of all (categorical) powers G^k of G remains the same as that of G, the adaptable chromatic number of G^k may increase with k. We conjecture that for all sufficiently large k the adaptable chromatic number of G^k equals the chromatic number of G. When G is complete, we prove this conjecture with k>=4, and offer additional evidence suggesting it may hold with k>=2. We also discuss other products and propose several open problems.

Linear Generalized Polynomials with Finiteness Conditions

Abstract Let R be a semiprime algebra over a field F, and let I be either an ideal of R or a right ideal with zero left annihilator in R. Let and , where a i , b i ∈ R. Suppose that dim F B 1 < ∞ (|B 1| < ∞). Then dim F B 2 < ∞ and dim F B 1 = dim F B 2 (resp. |B 2| < ∞ and |B 1| = |B 2|).