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In this paper, we discuss parameterized algorithms for variants of the partial vertex cover problem. Recall that in the classical vertex cover problem (VC), we are given a graph \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G = (V, E)$$\end{document} and a number K and asked if we can cover all of the edges in E, using at most K vertices from V. In the partial vertex cover problem (PVC), in addition to the parameter K, we are given a second parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K'$$\end{document} and the question is whether we can cover at least \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K'$$\end{document} of the edges in E using at most K vertices from V. The weighted generalizations of the VC and PVC problems are called the weighted vertex cover (WVC) and the partial weighted vertex cover problem (WPVC) respectively. In the WPVC problem, we are given two parameters R and L, associated respectively with the vertex set V and edge set E of the graph \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{G}$$\end{document}. Additionally, we are given non-negative integral weight functions for the vertices and the edges. The goal then is to cover edges of total weight at least L, using vertices of total weight at most R. (In the WVC problem, the goal is to cover all the edges with vertices whose total weight is at most R). This paper studies several variants of the PVC problem and establishes new results from the perspective of fixed-parameter tractability and W[1]-hardness. We also introduce a new problem called the partial vertex cover with matching constraint and show that it is fixed-parameter tractable for a certain class of graphs.

In the classical partial vertex cover problem, we are given a graph $G$ and two positive integers $R$ and $L$. The goal is to check whether there is a subset $V'$ of $V$ of size at most $R$, such that $V'$ covers at least $L$ edges of $G$. The problem is NP-hard as it includes the Vertex Cover problem. Previous research has addressed the extension of this problem where one has weight-functions defined on sets of vertices and edges of $G$. In this paper, we consider the following version of the problem where on the input we are given an edge-weighted bipartite graph $G$, and three positive integers $R$, $S$ and $T$. The goal is to check whether $G$ has a subset $V'$ of vertices of $G$ of size at most $R$, such that the edges of $G$ covered by $V'$ have weight at least $S$ and they include a matching of weight at least $T$. In the paper, we address this problem from the perspective of fixed-parameter tractability. One of our hardness results is obtained via a reduction from the bi-objective knapsack problem, which we show to be W[1]-hard with respect to one of parameters. We believe that this problem might be useful in obtaining similar results in other situations., 18 pages

A function $f:N\rightarrow N$ is sublinear, if \[\lim_{x\rightarrow +\infty}\frac{f(x)}{x}=0.\] If $A$ is an Abelian group, $G$ is a graph and $\phi$ is an $A$-flow in $G$, then let $N(\phi)$ be the nullity of $\phi$, that is, the set of edges $e$ of $G$ with $\phi(e)=0$. In this paper we show that (a) Tutte's 5-flow conjecture is equivalent to the statement that there is a sublinear function $f$, such that all $3$-edge-connected cubic graphs admit a $\mathbb{Z}_5$-flow $\phi$ (not necessarily no-where zero), such that $|N(\phi)|\leq f(|E(G)|)$; (b) Tutte's 4-flow conjecture is equivalent to the statement that there is a sublinear function $f$, such that all bridgeless graphs without a Petersen minor admit a $\mathbb{Z}_4$-flow $\phi$ (not necessarily no-where zero), such that $|N(\phi)|\leq f(|E(G)|)$; (c) Tutte's 3-flow conjecture is equivalent to the statement that there is a sublinear function $f$, such that all $4$-edge-connected graphs admit a $\mathbb{Z}_3$-flow $\phi$ (not necessarily no-where zero), such that $|N(\phi)|\leq f(|E(G)|)$., Comment: the results of the preprint have been obtained previously in "M. Kochol, Equivalences between hamiltonicity and flow conjectures, and the sublinear defect property, Discrete Mathematics 254 (2002) 221 -- 230", https://doi.org/10.1016/S0012-365X(01)00372-7

Many conjectures and open problems in graph theory can either be reduced to cubic graphs or are directly stated for cubic graphs. Furthermore, it is known that for a lot of problems, a counterexample must be a snark, i.e. a bridgeless cubic graph which is not 3--edge-colourable. In this paper we deal with the fact that the family of potential counterexamples to many interesting conjectures can be narrowed even further to the family ${\cal S}_{\geq 5}$ of bridgeless cubic graphs whose edge set cannot be covered with four perfect matchings. The Cycle Double Cover Conjecture, the Shortest Cycle Cover Conjecture and the Fan-Raspaud Conjecture are examples of statements for which ${\cal S}_{\geq 5}$ is crucial. In this paper, we study parameters which have the potential to further refine ${\cal S}_{\geq 5}$ and thus enlarge the set of cubic graphs for which the mentioned conjectures can be verified. We show that ${\cal S}_{\geq 5}$ can be naturally decomposed into subsets with increasing complexity, thereby producing a natural scale for proving these conjectures. More precisely, we consider the following parameters and questions: given a bridgeless cubic graph, (i) how many perfect matchings need to be added, (ii) how many copies of the same perfect matching need to be added, and (iii) how many 2--factors need to be added so that the resulting regular graph is Class I? We present new results for these parameters and we also establish some strong relations between these problems and some long-standing conjectures., 27 pages, 16 figures

An equitable coloring of a graph $G$ is a proper vertex coloring of $G$ such that the sizes of any two color classes differ by at most one. In the paper, we pose a conjecture that offers a gap-one bound for the smallest number of colors needed to equitably color every block graph. In other words, the difference between the upper and the lower bounds of our conjecture is at most one. Thus, in some sense, the situation is similar to that of chromatic index, where we have the classical theorem of Vizing and the Goldberg conjecture for multigraphs. The results obtained in the paper support our conjecture. More precisely, we verify it in the class of well-covered block graphs, which are block graphs in which each vertex belongs to a maximum independent set. We also show that the conjecture is true for block graphs, which contain a vertex that does not lie in an independent set of size larger than two. Finally, we verify the conjecture for some symmetric-like block graphs. In order to derive our results we obtain structural characterizations of block graphs from these classes., 21 pages, 12 figures

In the weighted partial vertex cover problem (WPVC), we are given a graph $G=(V,E)$, cost function $c:V\rightarrow N$, profit function $p:E\rightarrow N$, and positive integers $R$ and $L$. The goal is to check whether there is a subset $V'\subseteq V$ of cost at most $R$, such that the total profit of edges covered by $V'$ is at least $L$. In this paper we study the fixed-parameter tractability of WPVC in bipartite graphs (WPVCB). By extending the methods of Amini et al., we show that WPVCB is FPT with respect to $R$ if $c\equiv 1$. On the negative side, it is $W[1]$-hard for arbitrary $c$, even when $p\equiv 1$. In particular, WPVCB is $W[1]$-hard parameterized by $R$. We complement this negative result by proving that for bounded-degree graphs WPVC is FPT with respect to $R$. The same result holds for the case of WPVCB when we allow to take only one fractional vertex. Additionally, we show that WPVC is FPT with respect to $L$. Finally, we discuss a variant of PVCB in which the edges covered are constrained to include a matching of prescribed size and derive a paramterized algorithm for the same., 12 pages, no figures

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., 31 pages, 3 figures

If ?$G$? and ?$H$? are two cubic graphs, then an ?$H$?-coloring of ?$G$? is a proper edge-coloring ?$f$? with the edges of ?$H$?, such that for each vertex ?$x$? of ?$G$?, there is a vertex ?$y$? of ?$H$? with ?$f(\partial_G(x)) = \partial_H(y)$?. If ?$G$? admits an ?$H$?-coloring, then we will write ?$H\prec G$?. The Petersen coloring conjecture of Jaeger (?$P_{10}$?-conjecture) states that for any bridgeless cubic graph ?$G$?, one has: ?$P_{10} \prec G$?. The ?$S_{10}$?-conjecture states that for any cubic graph ?$G$?, ?$S_{10} \prec G$?. In this paper, we introduce two new conjectures that are related to these conjectures. The first of them states that any cubic graph with a perfect matching admits an ?$S_{12}$?-coloring. The second one states that any cubic graph ?$G$? whose edge-set can be covered with four perfect matchings, admits a ?$P_{12}$?-coloring. We call these new conjectures ?$S_{12}$?-conjecture and ?$P_{12}$?-conjecture, respectively. Our first results justify the choice of graphs in ?$S_{12}$?-conjecture and ?$P_{12}$?-conjecture. Next, we characterize the edges of ?$P_{12}$? that may be fictive in a ?$P_{12}$?-coloring of a cubic graph ?$G$?. Finally, we relate the new conjectures to the already known conjectures by proving that ?$S_{12}$?-conjecture implies ?$S_{10}$?-conjecture, and ?$P_{12}$?-conjecture and ?$(5, 2)$?-Cycle cover conjecture together imply ?$P_{10}$?-conjecture. Our main tool for proving the latter statement is a new reformulation of ?$(5, 2)$?-Cycle cover conjecture, which states that the edge-set of any claw-free bridgeless cubic graph can be covered with four perfect matchings. Če sta ?$G$? in ?$H$? kubična grafa, potem je ?$H$?-barvanje grafa ?$G$? pravilno povezavno barvanje ?$f$? s povezavami grafa ?$H$?, takšno da za vsako vozlišče ?$x$? grafa ?$G$? obstaja vozlišče ?$y$? grafa ?$H$?, za katero je ?$f(\partial_G(x)) = \partial_H(y)$?. Če ?$G$? dopušca ?$H$?-barvanje, potem bomo pisali ?$H\prec G$?. Jaegerjeva domneva o petersenskem barvanju (?$P_{10}$?-domneva) pravi, da za poljuben brezmostni kubični graf ?$G$? velja ?$P_{10} \prec G$?. ?$S_{10}$?-domneva pravi, da za poljuben kubični graf ?$G$? velja ?$S_{10} \prec G$?. V članku vpeljeva dve novi domnevi, ki sta povezani s tema domnevama. Prva od njiju pravi, da poljuben kubični graf s popolnim prirejanjem dopušca ?$S_{12}$?-barvanje. Druga pravi, da poljuben kubični graf ?$G$?, katerega povezavno množico se da pokriti s štirimi popolnimi prirejanji, dopušca ?$P_{12}$?-barvanje. Ti dve novi domnevi imenujeva ?$S_{12}$?-domneva in ?$P_{12}$?-domneva. Najin prvi rezultat opravičuje izbiro grafov v ?$S_{12}$?-domnevi in ?$P_{12}$?-domnevi. Nadalje, karakterizirava povezave v ?$P_{12}$?, ki lahko nastopajo v ?$P_{12}$?-barvanju kubičnega grafa ?$G$?. Nazadnje, poveževa novi domnevi z že znanimi domnevami, ko dokaževa, da ?$S_{12}$?-domneva implicira ?$S_{10}$?-domnevo, in da ?$P_{12}$?-domneva ter ?$(5, 2)$?-ciklična krovna domneva skupaj implicirata ?$P_{10}$?-domnevo. Najino glavno orodje za dokaz zadnje trditve je nova reformulacija ?$(5, 2)$?-ciklične krovne domneve, ki pravi, da se povezavna množica poljubnega brezmostnega kubičpnega grafa brez trizobov da pokriti s štirimi popolnimi prirejanji.

If $G$ and $H$ are two cubic graphs, then an $H$-coloring of $G$ is a proper edge-coloring $f$ with edges of $H$, such that for each vertex $x$ of $G$, there is a vertex $y$ of $H$ with $f(\partial_G(x))=\partial_H(y)$. If $G$ admits an $H$-coloring, then we will write $H\prec G$. The Petersen coloring conjecture of Jaeger ($P_{10}$-conjecture) states that for any bridgeless cubic graph $G$, one has: $P_{10}\prec G$. The Sylvester coloring conjecture ($S_{10}$-conjecture) states that for any cubic graph $G$, $S_{10}\prec G$. In this paper, we introduce two new conjectures that are related to these conjectures. The first of them states that any cubic graph with a perfect matching admits an $S_{12}$-coloring. The second one states that any cubic graph $G$ whose edge-set can be covered with four perfect matchings, admits a $P_{12}$-coloring. We call these new conjectures $S_{12}$-conjecture and $P_{12}$-conjecture, respectively. Our first results justify the choice of graphs in $S_{12}$-conjecture and $P_{12}$-conjecture. Next, we characterize the edges of $P_{12}$ that may be fictive in a $P_{12}$-coloring of a cubic graph $G$. Finally, we relate the new conjectures to the already known conjectures by proving that $S_{12}$-conjecture implies $S_{10}$-conjecture, and $P_{12}$-conjecture and $(5,2)$-Cycle cover conjecture together imply $P_{10}$-conjecture. Our main tool for proving the latter statement is a new reformulation of $(5,2)$-Cycle cover conjecture, which states that the edge-set of any claw-free bridgeless cubic graph can be covered with four perfect matchings., 15 pages, 6 figures

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 figures

If $G$ and $H$ are two cubic graphs, then an $H$-coloring of $G$ is a proper edge-coloring $f$ with edges of $H$, such that for each vertex $x$ of $G$, there is a vertex $y$ of $H$ with $f(\partial_G(x))=\partial_H(y)$. If $G$ admits an $H$-coloring, then we will write $H\prec G$. The Petersen coloring conjecture of Jaeger states that for any bridgeless cubic graph $G$, one has: $P\prec G$. The second author has recently introduced the Sylvester coloring conjecture, which states that for any cubic graph $G$ one has: $S\prec G$. Here $S$ is the Sylvester graph on $10$ vertices. In this paper, we prove the analogue of Sylvester coloring conjecture for cubic pseudo-graphs. Moreover, we show that if $G$ is any connected simple cubic graph $G$ with $G\prec P$, then $G = P$. This implies that the Petersen graph does not admit an $S_{16}$-coloring, where $S_{16}$ is the smallest connected simple cubic graph without a perfect matching. $S_{16}$ has $16$ vertices. %We conjecture that there are infinitely many connected cubic simple graphs which do not admit an %$S_{16}$-coloring. Finally, we obtain $2$ results towards the Sylvester coloring conjecture. The first result states that any cubic graph $G$ has a coloring with edges of Sylvester graph $S$ such that at least $\frac45$ of vertices of $G$ meet the conditions of Sylvester coloring conjecture. The second result states that any claw-free cubic graph graph admits an $S$-coloring. This results is an application of our result on cubic pseudo-graphs., 18 pages, 14 figures

This book is based on Graph Theory courses taught by P.A. Petrosyan, V.V. Mkrtchyan and R.R. Kamalian at Yerevan State University., 215 pages (in Armenian)

An $r$-graph is an $r$-regular graph where every odd set of vertices is connected by at least $r$ edges to the rest of the graph. Seymour conjectured that any $r$-graph is $r+1$-edge-colorable, and also that any $r$-graph contains $2r$ perfect matchings such that each edge belongs to two of them. We show that the minimum counter-example to either of these conjectures is a brick. Furthermore we disprove a variant of a conjecture of Fan, Raspaud., 4 pages

A graph $G$ is class II, if its chromatic index is at least $\Delta+1$. Let $H$ be a maximum $\Delta$-edge-colorable subgraph of $G$. The paper proves best possible lower bounds for $\frac{|E(H)|}{|E(G)|}$, and structural properties of maximum $\Delta$-edge-colorable subgraphs. It is shown that every set of vertex-disjoint cycles of a class II graph with $\Delta\geq3$ can be extended to a maximum $\Delta$-edge-colorable subgraph. Simple graphs have a maximum $\Delta$-edge-colorable subgraph such that the complement is a matching. Furthermore, a maximum $\Delta$-edge-colorable subgraph of a simple graph is always class I., Comment: 13 pages, 2 figures, the proof of the Lemma 1 is corrected

We show that if G is a connected bridgeless cubic graph whose every 2-factor is comprised of cycles of length five then G is the Petersen graph., 6 pages