Trees in tournaments

AbstractA digraph is said to be n-unavoidable if every tournament of order n contains it as a subgraph. Let f(n) be the smallest integer such that every oriented tree is f(n)-unavoidable. Sumner (see (Reid and Wormald, Studia Sci. Math. Hungaria 18 (1983) 377)) noted that f(n)⩾2n−2 and conjectured that equality holds. Häggkvist and Thomason established the upper bounds f(n)⩽12n and f(n)⩽(4+o(1))n. Let g(k) be the smallest integer such that every oriented tree of order n with k leaves is (n+g(k))-unavoidable. Häggkvist and Thomason (Combinatorica 11 (1991) 123) proved that g(k)⩽2512k3. Havet and Thomassé conjectured that g(k)⩽k−1. We study here the special case where the tree is a merging of paths (the union of disjoint paths emerging from a common origin). We prove that a merging of order n of k paths is (n+32(k2−3k)+5)-unavoidable. In particular, a tree with three leaves is (n+5)-unavoidable, i.e. g(3)⩽5. By studying trees with few leaves, we then prove that f(n)⩽385n−6

Trees in tournaments

AbstractA digraph is said to be n-unavoidable if every tournament of order n contains it as a subgraph. Let f(n) be the smallest integer such that every oriented tree is f(n)-unavoidable. Sumner (see (Reid and Wormald, Studia Sci. Math. Hungaria 18 (1983) 377)) noted that f(n)⩾2n−2 and conjectured that equality holds. Häggkvist and Thomason established the upper bounds f(n)⩽12n and f(n)⩽(4+o(1))n. Let g(k) be the smallest integer such that every oriented tree of order n with k leaves is (n+g(k))-unavoidable. Häggkvist and Thomason (Combinatorica 11 (1991) 123) proved that g(k)⩽2512k3. Havet and Thomassé conjectured that g(k)⩽k−1. We study here the special case where the tree is a merging of paths (the union of disjoint paths emerging from a common origin). We prove that a merging of order n of k paths is (n+32(k2−3k)+5)-unavoidable. In particular, a tree with three leaves is (n+5)-unavoidable, i.e. g(3)⩽5. By studying trees with few leaves, we then prove that f(n)⩽385n−6

Channel assignment and multicolouring of the induced subgraphs of the triangular lattice

AbstractA basic problem in the design of mobile telephone networks is to assign sets of radio frequency bands (colours) to transmitters (vertices) to avoid interference. Often the transmitters are laid out like vertices of a triangular lattice in the plane. We investigate the corresponding colouring problem of assigning sets of colours of given size k to vertices of the triangular lattice so that the sets of colours assigned to adjacent vertices are disjoint. We prove here that every triangle-free induced subgraph of the triangular lattice is ⌈7k/3⌉-[k]colourable. That means that it is possible to assign to each transmitter of such a network, k bands of a set of ⌈7k/3⌉, so that there is no interference

Finding an induced subdivision of a digraph

We consider the following problem for oriented graphs and digraphs: Given an oriented graph (digraph) $G$, does it contain an induced subdivision of a prescribed digraph $D$? The complexity of this problem depends on $D$ and on whether $G$ must be an oriented graph or is allowed to contain 2-cycles. We give a number of examples of polynomial instances as well as several NP-completeness proofs

Out-degree reducing partitions of digraphs

Let $k$ be a fixed integer. We determine the complexity of finding a $p$-partition $(V_1, \dots, V_p)$ of the vertex set of a given digraph such that the maximum out-degree of each of the digraphs induced by $V_i$, ($1\leq i\leq p$) is at least $k$ smaller than the maximum out-degree of $D$. We show that this problem is polynomial-time solvable when $p\geq 2k$ and ${\cal NP}$-complete otherwise. The result for $k=1$ and $p=2$ answers a question posed in \cite{bangTCS636}. We also determine, for all fixed non-negative integers $k_1,k_2,p$, the complexity of deciding whether a given digraph of maximum out-degree $p$ has a $2$-partition $(V_1,V_2)$ such that the digraph induced by $V_i$ has maximum out-degree at most $k_i$ for $i\in [2]$. It follows from this characterization that the problem of deciding whether a digraph has a 2-partition $(V_1,V_2)$ such that each vertex $v\in V_i$ has at least as many neighbours in the set $V_{3-i}$ as in $V_i$, for $i=1,2$ is ${\cal NP}$-complete. This solves a problem from \cite{kreutzerEJC24} on majority colourings.Comment: 11 pages, 1 figur

On the unavoidability of oriented trees

A digraph is {\it $n$-unavoidable} if it is contained in every tournament of order $n$. We first prove that every arborescence of order $n$ with $k$ leaves is $(n+k-1)$-unavoidable. We then prove that every oriented tree of order $n$ ($n\geq 2$) with $k$ leaves is $(\frac{3}{2}n+\frac{3}{2}k -2)$-unavoidable and $(\frac{9}{2}n -\frac{5}{2}k -\frac{9}{2})$-unavoidable, and thus $(\frac{21}{8} n- \frac{47}{16})$-unavoidable. Finally, we prove that every oriented tree of order $n$ with $k$ leaves is $(n+ 144k^2 - 280k + 124)$-unavoidable