1,989 research outputs found
Asymmetric coloring games on incomparability graphs
Consider the following game on a graph : Alice and Bob take turns coloring
the vertices of properly from a fixed set of colors; Alice wins when the
entire graph has been colored, while Bob wins when some uncolored vertices have
been left. The game chromatic number of is the minimum number of colors
that allows Alice to win the game. The game Grundy number of is defined
similarly except that the players color the vertices according to the first-fit
rule and they only decide on the order in which it is applied. The -game
chromatic and Grundy numbers are defined likewise except that Alice colors
vertices and Bob colors vertices in each round. We study the behavior of
these parameters for incomparability graphs of posets with bounded width. We
conjecture a complete characterization of the pairs for which the
-game chromatic and Grundy numbers are bounded in terms of the width of
the poset; we prove that it gives a necessary condition and provide some
evidence for its sufficiency. We also show that the game chromatic number is
not bounded in terms of the Grundy number, which answers a question of Havet
and Zhu
The Edge-Distinguishing Chromatic Number of Petal Graphs, Chorded Cycles, and Spider Graphs
The edge-distinguishing chromatic number (EDCN) of a graph is the minimum
positive integer such that there exists a vertex coloring
whose induced edge labels are
distinct for all edges . Previous work has determined the EDCN of paths,
cycles, and spider graphs with three legs. In this paper, we determine the EDCN
of petal graphs with two petals and a loop, cycles with one chord, and spider
graphs with four legs. These are achieved by graph embedding into looped
complete graphs.Comment: 23 pages, 1 figur
On retracts, absolute retracts, and folds in cographs
Let G and H be two cographs. We show that the problem to determine whether H
is a retract of G is NP-complete. We show that this problem is fixed-parameter
tractable when parameterized by the size of H. When restricted to the class of
threshold graphs or to the class of trivially perfect graphs, the problem
becomes tractable in polynomial time. The problem is also soluble when one
cograph is given as an induced subgraph of the other. We characterize absolute
retracts of cographs.Comment: 15 page
Group Sum Chromatic Number of Graphs
We investigate the \textit{group sum chromatic number} (\gchi(G)) of
graphs, i.e. the smallest value such that taking any Abelian group \gr of
order , there exists a function f:E(G)\rightarrow \gr such that the sums
of edge labels properly colour the vertices. It is known that
\gchi(G)\in\{\chi(G),\chi(G)+1\} for any graph with no component of order
less than and we characterize the graphs for which \gchi(G)=\chi(G).Comment: Accepted for publication in European Journal of Combinatorics,
Elsevie
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