Domination number of graphs with minimum degree five

We prove that for every graph $G$ on $n$ vertices and with minimum degree five, the domination number $\gamma(G)$ cannot exceed $n/3$. The proof combines an algorithmic approach and the discharging method. Using the same technique, we provide a shorter proof for the known upper bound $4n/11$ 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 $F$ there exist very dense graphs $G_n$ on $n$ vertices, namely with as many as ${n\choose 2}-cn$ edges for all $n$, for some constant $c=c(F)$, such that $G_n$ can be decomposed into edge-disjoint induced subgraphs isomorphic to~$F$. This result identifies and structurally explains a gap between the growth rates $O(n)$ and $\Omega(n^{3/2})$ on the minimum number of non-edges in graphs admitting an induced $F$-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 $(2:1)$ 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 $(a:b)$ biased game for $(a:b)\neq (2:1)$. 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

K3K_3-WORM colorings of graphs: Lower chromatic number and gaps in the chromatic spectrum

A $K_3$-WORM coloring of a graph $G$ is an assignment of colors to the vertices in such a way that the vertices of each $K_3$-subgraph of $G$ get precisely two colors. We study graphs $G$ 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 $K_3$-WORM coloring with two colors. In fact for every integer $k\ge 3$ there exists a $K_3$-WORM colorable graph in which the minimum number of colors is exactly $k$. There also exist $K_3$-WORM colorable graphs which have a $K_3$-WORM coloring with two colors and also with $k$ colors but no coloring with any of $3,\dots,k-1$ colors. We also prove that it is NP-hard to determine the minimum number of colors and NP-complete to decide $k$-colorability for every $k \ge 2$ (and remains intractable even for graphs of maximum degree 9 if $k=3$). On the other hand, we prove positive results for $d$-degenerate graphs with small $d$, 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

Approximability of the upper chromatic number of hypergraphs

A C-coloring of a hypergraph H = (X, E) is a vertex coloring φ : X → N such that each edge E ∈ E has at least two vertices with a common color. The related parameter over(χ, -) (H), called the upper chromatic number of H, is the maximum number of colors in a C-coloring of H. A hypertree is a hypergraph which has a host tree T such that each edge E ∈ E induces a connected subgraph in T. Notations n and m stand for the number of vertices and edges, respectively, in a generic input hypergraph. We establish guaranteed polynomial-time approximation ratios for the difference n - over(χ, -) (H), which is 2 + 2 ln (2 m) on hypergraphs in general, and 1 + ln m on hypertrees. The latter ratio is essentially tight as we show that n - over(χ, -) (H) cannot be approximated within (1 - ε{lunate}) ln m on hypertrees (unless NP⊆ DTIME(nO (log log n)) ). Furthermore, over(χ, -) (H) does not have O (n1 - ε{lunate})-approximation and cannot be approximated within additive error o (n) on the class of hypertrees (unless P = NP). © 2014 Elsevier B.V. All rights reserved