5 research outputs found
Disjoint Total Dominating Sets in Near-Triangulations
We show that every simple planar near-triangulation with minimum degree at
least three contains two disjoint total dominating sets. The class includes all
simple planar triangulations other than the triangle. This affirms a conjecture
of Goddard and Henning [Thoroughly dispersed colorings, J. Graph Theory, 88
(2018) 174-191]
Maker-Breaker total domination game
Maker-Breaker total domination game in graphs is introduced as a natural
counterpart to the Maker-Breaker domination game recently studied by Duch\^ene,
Gledel, Parreau, and Renault. Both games are instances of the combinatorial
Maker-Breaker games. The Maker-Breaker total domination game is played on a
graph by two players who alternately take turns choosing vertices of .
The first player, Dominator, selects a vertex in order to totally dominate
while the other player, Staller, forbids a vertex to Dominator in order to
prevent him to reach his goal.
It is shown that there are infinitely many connected cubic graphs in which
Staller wins and that no minimum degree condition is sufficient to guarantee
that Dominator wins when Staller starts the game. An amalgamation lemma is
established and used to determine the outcome of the game played on grids.
Cacti are also classified with respect to the outcome of the game. A connection
between the game and hypergraphs is established. It is proved that the game is
PSPACE-complete on split and bipartite graphs. Several problems and questions
are also posed.Comment: 21 pages, 5 figure
Polychromatic colorings of certain subgraphs of complete graphs and maximum densities of substructures of a hypercube
If G is a graph and H is a set of subgraphs of G, an edge-coloring of G is H-polychromatic if every graph from H gets all colors present in G on its edges. The H-polychromatic number of G, polyHG, is the largest number of colors in an H-polychromatic coloring. We determine polyHG exactly when G is a complete graph on n vertices, q a fixed nonnegative integer, and H is the family of one of: all matchings spanning n-q vertices, all 2-regular graphs spanning at least n-q vertices, or all cycles of length precisely n-q.
For H, K, subsets of the vertex set V(Qd) of the d-cube Qd, K is an exact copy of H if there is an automorphism of Qd sending H to K. For a positive integer, d, and a configuration in Qd, H, we define λ(H,d) as the limit as n goes to infinity of the maximum fraction, over all subsets S of V(Qn), of sub-d-cubes of Qn whose intersection with S is an exact copy of H.
We determine λ(C8,4) and λ(P4,3) where C8 is a “perfect” 8-cycle in Q4 and P4 is a “perfect” path with 4 vertices in Q3, λ(H,d) for several configurations in Q2, Q3, and Q4, and an infinite family of configurations.
Strong connections exist with extensions Ramsey numbers for cycles in a graph, counting sequences with certain properties, inducibility of graphs, and we determine the inducibility of two vertex disjoint edges in the family of bipartite graphs