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Julien Portier

Publications and source records attributed to Julien Portier.

28 records · Page 2Linked to original sources
arXiv2023

The complexity of decomposing a graph into a matching and a bounded linear forest

Deciding whether a graph can be edge-decomposed into a matching and a $k$-bounded linear forest was recently shown by Campbell, H{ö}rsch and Moore to be NP-complete for every $k \ge 9$, and solvable in polynomial time for $k=1,2$. In the first part of this paper, we close this gap by showing that this problem is in NP-complete for every $k \ge 3$. In the second part of the paper, we show that deciding whether a graph can be edge-decomposed into a matching and a $k$-bounded star forest is polynomially solvable for any $k \in \mathbb{N} \cup \{ \infty \}$, answering another question by Campbell, H{ö}rsch and Moore from the same paper.

cs.CC↗
arXiv2023

Bounds for the Competition-Independence game on trees

In this paper we prove that Sweller has a strategy so that the Sweller-Start Competition-Independence game lasts at least $(5n+3)/13$ moves for every tree. Moreover, we show that there exist arbitrarily large trees such that the Sweller-Start Competition-Independence game lasts at most $(5n+26)/12$ moves, disproving a conjecture by Henning.

math.CO↗
arXiv2023

Progress towards the 1/2-Conjecture for the domination game

The domination game is played on a graph $G$ by two players, Dominator and Staller, who alternate in selecting vertices until each vertex in the graph $G$ is contained in the closed neighbourhood of the set of selected vertices. Dominator's aim is to reach this state in as few moves as possible, whereas Staller wants the game to last as long as possible. In this paper, we prove that if $G$ has $n$ vertices and minimum degree at least 2, then Dominator has a strategy to finish the domination game on $G$ within $10n/17+1/17$ moves, thus making progress towards a conjecture by Bujt{á}s, Iršič and Klavžar.

math.CO↗
arXiv2022

A proof of the 3/4 conjecture for the total domination game

In this paper we describe a strategy for Dominator that finishes the total domination game in at most $3/4n$ moves for every graph $G$ on $n$ vertices without any isolated vertices or edges, confirming the 3/4-conjecture for the total domination game made by Henning, Klav{ž}ar, and Rall.

math.CO↗
arXiv2022

On the number of minimum dominating sets and total dominating sets in forests

We show that the maximum number of minimum dominating sets of a forest with domination number $γ$ is at most $\sqrt{5}^γ$ and construct for each $γ$ a tree with domination number $γ$ that has more than $\frac{2}{5}\sqrt{5}^γ$ minimum dominating sets. Furthermore, we disprove a conjecture about the number of minimum total dominating sets in forests by Henning, Mohr and Rautenbach.

math.CO↗
arXiv2022

Ternary Egyptian fractions with prime denominator

For a prime number $p$, let $A_3(p)= | \{ m \in \mathbb{N}: \exists m_1,m_2,m_3 \in \mathbb{N}, \frac{m}{p}=\frac{1}{m_1}+\frac{1}{m_2}+\frac{1}{m_3} \} |$. In 2019 Luca and Pappalardi proved that $x (\log x)^3 \ll \sum_{p \le x} A_{3}(p) \ll x (\log x)^5$. We improve the upper bound, showing $\sum_{p \le x} A_{3}(p) \ll x (\log x)^3 (\log \log x)^2$.

math.NT↗
arXiv2022

The odd chromatic number of a planar graph is at most 8

Petruševski and Škrekovski \cite{odd9} recently introduced the notion of an odd colouring of a graph: a proper vertex colouring of a graph $G$ is said to be \emph{odd} if for each non-isolated vertex $x \in V(G)$ there exists a colour $c$ appearing an odd number of times in $N(x)$. Petruševski and Škrekovski proved that for any planar graph $G$ there is an odd colouring using at most $9$ colours and, together with Caro \cite{oddremarks}, showed that $8$ colours are enough for a significant family of planar graphs. We show that $8$ colours suffice for all planar graphs.

math.CO↗
arXiv2021

A faster algorithm for Cops and Robbers

We present an algorithm of time complexity $O(kn^{k+2})$ deciding whether a graph $G$ on $n$ vertices is $k$-copwin. The fastest algorithm thus far had time complexity $O(n^{2k+2})$.

cs.DS↗
arXiv2021

A new lower bound on the pebbling number of the grid

A pebbling move on a graph consists of removing $2$ pebbles from a vertex and adding $1$ pebble to one of the neighbouring vertices. A vertex is called reachable if we can put $1$ pebble on it after a sequence of moves. The optimal pebbling number of a graph is the minimum number $m$ such that there exists a distribution of $m$ pebbles so that each vertex is reachable. For the case of a square grid $n \times m$, Győri, Katona and Papp recently showed that its optimal pebbling number is at least $\frac{2}{13}nm \approx 0.1538nm$ and at most $\frac{2}{7}nm +O(n+m) \approx 0.2857nm$. We improve the lower bound to $\frac{5092}{28593}nm +O(m+n) \approx 0.1781nm$.

math.CO↗