On maximum kk-edge-colorable subgraphs of bipartite graphs

If $k\geq 0$, then a $k$-edge-coloring of a graph $G$ is an assignment of colors to edges of $G$ from the set of $k$ colors, so that adjacent edges receive different colors. A $k$-edge-colorable subgraph of $G$ is maximum if it is the largest among all $k$-edge-colorable subgraphs of $G$. For a graph $G$ and $k\geq 0$, let $\nu_{k}(G)$ be the number of edges of a maximum $k$-edge-colorable subgraph of $G$. In 2010 Mkrtchyan et al. proved that if $G$ is a cubic graph, then $\nu_2(G)\leq \frac{|V|+2\nu_3(G)}{4}$. This result implies that if the cubic graph $G$ contains a perfect matching, in particular when it is bridgeless, then $\nu_2(G)\leq \frac{\nu_1(G)+\nu_3(G)}{2}$. One may wonder whether there are other interesting graph-classes, where a relation between $\nu_2(G)$ and $\frac{\nu_1(G)+\nu_3(G)}{2}$ can be proved. Related with this question, in this paper we show that $\nu_{k}(G) \geq \frac{\nu_{k-i}(G) + \nu_{k+i}(G)}{2}$ for any bipartite graph $G$, $k\geq 0$ and $i=0,1,...,k$.Comment: 11 pages, 1 figur

Assigning tasks to agents under time conflicts: a parameterized complexity approach

We consider the problem of assigning tasks to agents under time conflicts, with applications also to frequency allocations in point-to-point wireless networks. In particular, we are given a set $V$ of $n$ agents, a set $E$ of $m$ tasks, and $k$ different time slots. Each task can be carried out in one of the $k$ predefined time slots, and can be represented by the subset $e\subseteq E$ of the involved agents. Since each agent cannot participate to more than one task simultaneously, we must find an allocation that assigns non-overlapping tasks to each time slot. Being the number of slots limited by $k$, in general it is not possible to executed all the possible tasks, and our aim is to determine a solution maximizing the overall social welfare, that is the number of executed tasks. We focus on the restriction of this problem in which the number of time slots is fixed to be $k=2$, and each task is performed by exactly two agents, that is $|e|=2$. In fact, even under this assumptions, the problem is still challenging, as it remains computationally difficult. We provide parameterized complexity results with respect to several reasonable parameters, showing for the different cases that the problem is fixed-parameter tractable or it is paraNP-hard.Comment: 31 pages, 3 figure