Relaxed Game Chromatic Numbers of Complete Multipartite Graphs

Competitive graph coloring is investigated by studying a game with two players, Alice and Bob, on a finite graph G with a set of r colors. Alice and Bob alternately color the vertices of G with legal colors. In the k-relaxed coloring game, a color c is legal for a vertex v if v has at most k neighbors previously colored c. New results about the 0, 1, and 2-relaxed game chromatic numbers will be presented, completely classifying the 0 and 1-relaxed games and partially classifying the 2-relaxed game. These results will be presented in the context of previous research and given a direction of where they need to go next

The Relaxed Game Chromatic Index of \u3cem\u3ek\u3c/em\u3e-Degenerate Graphs

The (r, d)-relaxed coloring game is a two-player game played on the vertex set of a graph G. We consider a natural analogue to this game on the edge set of G called the (r, d)-relaxed edge-coloring game. We consider this game on trees and more generally, on k-degenerate graphs. We show that if G is k-degenerate with ∆(G) = ∆, then the first player, Alice, has a winning strategy for this game with r = ∆+k−1 and d≥2k2 + 4k

1-Relaxed Edge-Sum Labeling Game

We introduce a new graph labeling and derive a game on graphs called the 1-relaxed modular edge-sum labeling game. Given a graph G and a natural number n, we define a labeling by assigning to each edge a number from {1,..., n} and assign a corresponding label for each vertex u by the sum of the labels of the edges incident to u, computing this sum modulo n. Similar to the chromatic number, we define L(G) for a graph G as the smallest n such that G has a proper labeling. We provide bounds for L(G) for various classes of graphs. Motivated by competitive graph coloring, we define a game on using modular edge-sum labeling and determine the chromatic game number for various classes of graphs. We will emphasize some characteristics that distinguish this labeling from traditional vertex coloring

Covering the complete graph by partitions

AbstractA (D, c)-coloring of the complete graph Kn is a coloring of the edges with c colors such that all monochromatic connected subgraphs have at most D vertices. Resolvable block designs with c parallel classes and with block size D are natural examples of (D, c)-colorings. However, (D, c)-colorings are more relaxed structures. We investigate the largest n such that Kn has a (D, c)-coloring. Our main tool is the fractional matching theory of hypergraphs