A lower bound on the acyclic matching number of subcubic graphs

The acyclic matching number of a graph $G$ is the largest size of an acyclic matching in $G$, that is, a matching $M$ in $G$ such that the subgraph of $G$ induced by the vertices incident to an edge in $M$ is a forest. We show that the acyclic matching number of a connected subcubic graph $G$ with $m$ edges is at least $m/6$ except for two small exceptions

Largest 2-regular subgraphs in 3-regular graphs

For a graph $G$, let $f_2(G)$ denote the largest number of vertices in a $2$-regular subgraph of $G$. We determine the minimum of $f_2(G)$ over $3$-regular $n$-vertex simple graphs $G$. To do this, we prove that every $3$-regular multigraph with exactly $c$ cut-edges has a $2$-regular subgraph that omits at most $\max\{0,\lfloor (c-1)/2\rfloor\}$ vertices. More generally, every $n$-vertex multigraph with maximum degree $3$ and $m$ edges has a $2$-regular subgraph that omits at most $\max\{0,\lfloor (3n-2m+c-1)/2\rfloor\}$ vertices. These bounds are sharp; we describe the extremal multigraphs

On strongly spanning kk-edge-colorable subgraphs

A subgraph $H$ of a multigraph $G$ is called strongly spanning, if any vertex of $G$ is not isolated in $H$, while it is called maximum $k$-edge-colorable, if $H$ is proper $k$-edge-colorable and has the largest size. We introduce a graph-parameter $sp(G)$, that coincides with the smallest $k$ that a graph $G$ has a strongly spanning maximum $k$-edge-colorable subgraph. Our first result offers some alternative definitions of $sp(G)$. Next, we show that $\Delta(G)$ is an upper bound for $sp(G)$, and then we characterize the class of graphs $G$ that satisfy $sp(G)=\Delta(G)$. Finally, we prove some bounds for $sp(G)$ that involve well-known graph-theoretic parameters.Comment: 12 pages, no figure

Graphs, Disjoint Matchings and Some Inequalities

For $k \geq 1$ and a graph $G$ let $\nu_k(G)$ denote the size of a maximum $k$-edge-colorable subgraph of $G$. Mkrtchyan, Petrosyan and Vardanyan proved that $\nu_2(G)\geq \frac45\cdot |V(G)|$, $\nu_3(G)\geq \frac76\cdot |V(G)|$ for any cubic graph $G$ ~\cite{samvel:2010}. They were also able to show that if $G$ is a cubic graph, then $\nu_2(G)+\nu_3(G)\geq 2\cdot |V(G)|$ ~\cite{samvel:2014} and $\nu_2(G) \leq \frac{|V(G)| + 2\cdot \nu_3(G)}{4}$ ~\cite{samvel:2010}. In the first part of the present work, we show that the last two inequalities imply the first two of them. Moreover, we show that $\nu_2(G) \geq \alpha \cdot \frac{|V(G)| + 2\cdot \nu_3(G)}{4}$, where $\alpha=\frac{16}{17}$, if $G$ is a cubic graph, $\alpha=\frac{20}{21}$, if $G$ is a cubic graph containing a perfect matching, $\alpha=\frac{44}{45}$, if $G$ is a bridgeless cubic graph. We also investigate the parameters $\nu_2(G)$ and $\nu_3(G)$ in the class of claw-free cubic graphs. We improve the lower bounds for $\nu_2(G)$ and $\nu_3(G)$ for claw-free bridgeless cubic graphs to $\nu_2(G)\geq \frac{35}{36}\cdot |V(G)|$ ($n \geq 48$), $\nu_3(G)\geq \frac{43}{45}\cdot |E(G)|$. On the basis of these inequalities we are able to improve the coefficient $\alpha$ for bridgeless claw-free cubic graphs. In the second part of the work, we prove lower bounds for $\nu_k(G)$ in terms of $\frac{\nu_{k-1}(G)+\nu_{k+1}(G)}{2}$ for $k\geq 2$ and graphs $G$ containing at most $1$ cycle. We also present the corresponding conjectures for bipartite and nearly bipartite graphs.Comment: 22 pages, 14 figure

Game matching number of graphs

We study a competitive optimization version of $\alpha'(G)$, the maximum size of a matching in a graph $G$. Players alternate adding edges of $G$ to a matching until it becomes a maximal matching. One player (Max) wants that matching to be large; the other (Min) wants it to be small. The resulting sizes under optimal play when Max or Min starts are denoted \Max(G) and \Min(G), respectively. We show that always |\Max(G)-\Min(G)|\le 1. We obtain a sufficient condition for \Max(G)=\alpha'(G) that is preserved under cartesian product. In general, \Max(G)\ge \frac23\alpha'(G), with equality for many split graphs, while \Max(G)\ge\frac34\alpha'(G) when $G$ is a forest. Whenever $G$ is a 3-regular $n$-vertex connected graph, \Max(G) \ge n/3, and there are such examples with \Max(G)\le 7n/18. For an $n$-vertex path or cycle, the answer is roughly $n/7$.Comment: 16 pages; this version improves explanations at a number of points and adds a few more example

Some bounds on the uniquely restricted matching number

A matching in a graph is uniquely restricted if no other matching covers exactly the same set of vertices. We establish tight lower bounds on the maximum size of a uniquely restricted matching in terms of order, size, and maximum degree

Directed Domination in Oriented Graphs

A directed dominating set in a directed graph $D$ is a set $S$ of vertices of $V$ such that every vertex $u \in V(D) \setminus S$ has an adjacent vertex $v$ in $S$ with $v$ directed to $u$. The directed domination number of $D$, denoted by $\gamma(D)$, is the minimum cardinality of a directed dominating set in $D$. The directed domination number of a graph $G$, denoted $\Gamma_d(G)$, which is the maximum directed domination number $\gamma(D)$ over all orientations $D$ of $G$. The directed domination number of a complete graph was first studied by Erd\"{o}s [Math. Gaz. 47 (1963), 220--222], albeit in disguised form. We extend this notion to directed domination of all graphs. If $\alpha$ denotes the independence number of a graph $G$, we show that if $G$ is a bipartite graph, we show that $\Gamma_d(G) = \alpha$. We present several lower and upper bounds on the directed domination number.Comment: 18 page

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

A tight lower bound on the matching number of graphs via Laplacian eigenvalues

Let $\alpha'$ and $\mu_i$ denote the matching number of a non-empty simple graph $G$ with $n$ vertices and the $i$-th smallest eigenvalue of its Laplacian matrix, respectively. In this paper, we prove a tight lower bound $$\alpha' \ge \min\left\{\Big\lceil\frac{\mu_2}{\mu_n} (n -1)\Big\rceil,\ \ \Big\lceil\frac{1}{2}(n-1)\Big\rceil \right\}.$$ This bound strengthens the result of Brouwer and Haemers who proved that if $n$ is even and $2\mu_2 \ge \mu_n$, then $G$ has a perfect matching. A graph $G$ is factor-critical if for every vertex $v\in V(G)$, $G-v$ has a perfect matching. We also prove an analogue to the result of Brouwer and Haemers mentioned above by showing that if $n$ is odd and $2\mu_2 \ge \mu_n$, then $G$ is factor-critical. We use the separation inequality of Haemers to get a useful lemma, which is the key idea in the proofs. This lemma is of its own interest and has other applications. In particular, we prove similar results for the number of balloons, spanning even subgraphs, as well as spanning trees with bounded degree.Comment: The first manuscript was done in May 2020, and the current manuscript was accepted by European Journal of Combinatorics in October 202

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