89 research outputs found
Domination number of graphs with minimum degree five
We prove that for every graph on vertices and with minimum degree
five, the domination number cannot exceed . The proof combines
an algorithmic approach and the discharging method. Using the same technique,
we provide a shorter proof for the known upper bound on the domination
number of graphs of minimum degree four.Comment: 17 page
Transversal designs and induced decompositions of graphs
We prove that for every complete multipartite graph there exist very
dense graphs on vertices, namely with as many as
edges for all , for some constant , such that can be
decomposed into edge-disjoint induced subgraphs isomorphic to~. This result
identifies and structurally explains a gap between the growth rates and
on the minimum number of non-edges in graphs admitting an
induced -decomposition
The Disjoint Domination Game
We introduce and study a Maker-Breaker type game in which the issue is to
create or avoid two disjoint dominating sets in graphs without isolated
vertices. We prove that the maker has a winning strategy on all connected
graphs if the game is started by the breaker. This implies the same in the
biased game also in the maker-start game. It remains open to
characterize the maker-win graphs in the maker-start non-biased game, and to
analyze the biased game for . For a more restricted
variant of the non-biased game we prove that the maker can win on every graph
without isolated vertices.Comment: 18 page
-WORM colorings of graphs: Lower chromatic number and gaps in the chromatic spectrum
A -WORM coloring of a graph is an assignment of colors to the
vertices in such a way that the vertices of each -subgraph of get
precisely two colors. We study graphs which admit at least one such
coloring. We disprove a conjecture of Goddard et al. [Congr. Numer., 219 (2014)
161--173] who asked whether every such graph has a -WORM coloring with two
colors. In fact for every integer there exists a -WORM colorable
graph in which the minimum number of colors is exactly . There also exist
-WORM colorable graphs which have a -WORM coloring with two colors
and also with colors but no coloring with any of colors. We
also prove that it is NP-hard to determine the minimum number of colors and
NP-complete to decide -colorability for every (and remains
intractable even for graphs of maximum degree 9 if ). On the other hand,
we prove positive results for -degenerate graphs with small , also
including planar graphs. Moreover we point out a fundamental connection with
the theory of the colorings of mixed hypergraphs. We list many open problems at
the end.Comment: 18 page
F-WORM colorings: Results for 2-connected graphs
Given two graphs F and G, an F-WORM coloring of G is an assignment of colors to its vertices in such a way that no F-subgraph of G is monochromatic or rainbow. If G has at least one such coloring, then it is called F-WORM colorable and W−(G,F) denotes the minimum possible number of colors. Here, we consider F-WORM colorings with a fixed 2-connected graph F and prove the following three main results: (1) For every natural number k, there exists a graph G which is F-WORM colorable and W−(G,F)=k; (2) It is NP-complete to decide whether a graph is F-WORM colorable; (3) For each k≥|V(F)|−1, it is NP-complete to decide whether a graph G satisfies W−(G,F)≤k. This remains valid on the class of F-WORM colorable graphs of bounded maximum degree. We also prove that for each n≥3, there exists a graph G and integers r and s such that s≥r+2, G has Kn-WORM colorings with exactly r and also with s colors, but it admits no Kn-WORM colorings with exactly r+1,…,s−1 colors. Moreover, the difference s−r can be arbitrarily large. © 2017 Elsevier B.V
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