This paper investigates a zero-sum game played on a weighted connected graph G between two players, the tree player and the edge player. At each play, the tree player chooses a spanning tree T and the edge player chooses an edge e. The payo to the edge player
In the domination game on a graph G, two players called Dominator and Staller alternately select vertices of G. Each vertex chosen must strictly increase the number of vertices dominated; the game ends when the chosen set becomes a dominating set of G. Dominator aims to minimize the size of the resulting dominating set, while Staller aims to maximize it. When both players play optimally, the size of the dominating set produced is the game domination number of G, denoted by γ g (G) when Dominator plays first and by γ ′ g (G) when Staller plays first. We prove that γ g (G) ≤ 7n/11 when G is an isolate-free n-vertex forest and that γ g (G) ≤ ⌈7n/10⌉ for any isolate-free n-vertex graph. In both cases we conjecture that γ g (G) ≤ 3n/5 and prove it when G is a forest of nontrivial caterpillars. We also resolve conjectures of Brešar, Klavžar, and Rall by showing that always γ ′ g (G) ≤ γ g (G) + 1, that for k ≥ 2 there are graphs G satisfying γ g (G) = 2k and γ ′ g (G) = 2k − 1, and thatwhen G is a forest. Our results follow from fundamental lemmas about the domination game that simplify its analysis and may be useful in future research.
connected graph having large minimum vertexdegree must have a spanning tree with manyleaves. In particular,let l(n, k) e the maximum integer m such that every connected n-vertexgraph with minimum degree at least k has a spanning tree with at least m leaves. Then l(n,3) ≥ n/4 + 2, l(n,4) ≥ (2n + 8) /5, and l(n, k) ≤ n − 3n/(k + 1)+2f or all k.T he lower bounds are rovedb ya na lgorithm that constructs a spanning tree with at least the desired number of leaves. Finally, l(n, k) ≥ (1 − blnk/k)n for large k,again provedalgorithmically,where b is anyconstant exceeding 2.5.
ABSTRACT. The Borsuk-Ulam theorem of topology is applied to a problem in discrete mathematics.A bisection of a necklace with k colors of beads is a collection of intervals whose union captures half the beads of each color. Every necklace with fc colors has a bisection formed by at most k cuts. Higherdimensional generalizations are considered.
Given lists of available colors assigned to the vertices of a graph G, a list coloring is a proper coloring of G such that the color on each vertex is chosen from its list. If the lists all have size k, then a list coloring is equitable if each color appears on at most dn(G )/ke vertices. A graph is equitably k-choosable if such a coloring exists whenever the lists all have size k. We prove that G is equitably k-choosable when k ! maxfÁ(G ),n(G )/ ------------------
Kostochka and West
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.