39 research outputs found
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
A relative of Hadwiger's conjecture
Hadwiger's conjecture asserts that if a simple graph has no
minor, then its vertex set can be partitioned into stable sets. This
is still open, but we prove under the same hypotheses that can be
partitioned into sets , such that for , the
subgraph induced on has maximum degree at most a function of . This is
sharp, in that the conclusion becomes false if we ask for a partition into
sets with the same property.Comment: 6 page
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
The t-improper chromatic number of random graphs
We consider the -improper chromatic number of the Erd{\H o}s-R{\'e}nyi
random graph . The t-improper chromatic number of is
the smallest number of colours needed in a colouring of the vertices in which
each colour class induces a subgraph of maximum degree at most . If ,
then this is the usual notion of proper colouring. When the edge probability
is constant, we provide a detailed description of the asymptotic behaviour
of over the range of choices for the growth of .Comment: 12 page
Clique-Relaxed Graph Coloring
We define a generalization of the chromatic number of a graph G called the k-clique-relaxed chromatic number, denoted χ(k)(G). We prove bounds on χ(k)(G) for all graphs G, including corollaries for outerplanar and planar graphs. We also define the k-clique-relaxed game chromatic number, χg(k)(G), of a graph G. We prove χg(2)(G)≤ 4 for all outerplanar graphs G, and give an example of an outerplanar graph H with χg(2)(H) ≥ 3. Finally, we prove that if H is a member of a particular subclass of outerplanar graphs, then χg(2)(H) ≤ 3
The Relaxed Edge-Coloring Game and \u3cem\u3ek\u3c/em\u3e-Degenerate Graphs
The (r, d)-relaxed edge-coloring game is a two-player game using r colors played on the edge set of a graph G. We consider this game on forests and more generally, on k-degenerate graphs. If F is a forest with ∆(F) = ∆, then the first player, Alice, has a winning strategy for this game with r = ∆ − j and d ≥ 2j + 2 for 0 ≤ j ≤ ∆ − 1. This both improves and generalizes the result for trees in [10]. More broadly, we generalize the main result in [10] by showing that if G is k-degenerate with ∆(G) = ∆ and j ∈ [∆ + k − 1], then there exists a function h(k, j) such that Alice has a winning strategy for this game with r = ∆ + k − j and d ≥ h(k, j)