Approximation bounds on maximum edge 2-coloring of dense graphs

For a graph $G$ and integer $q\geq 2$, an edge $q$-coloring of $G$ is an assignment of colors to edges of $G$, such that edges incident on a vertex span at most $q$ distinct colors. The maximum edge $q$-coloring problem seeks to maximize the number of colors in an edge $q$-coloring of a graph $G$. The problem has been studied in combinatorics in the context of {\em anti-Ramsey} numbers. Algorithmically, the problem is NP-Hard for $q\geq 2$ and assuming the unique games conjecture, it cannot be approximated in polynomial time to a factor less than $1+1/q$. The case $q=2$, is particularly relevant in practice, and has been well studied from the view point of approximation algorithms. A $2$-factor algorithm is known for general graphs, and recently a $5/3$-factor approximation bound was shown for graphs with perfect matching. The algorithm (which we refer to as the matching based algorithm) is as follows: "Find a maximum matching $M$ of $G$. Give distinct colors to the edges of $M$. Let $C_1,C_2,\ldots, C_t$ be the connected components that results when M is removed from G. To all edges of $C_i$ give the $(|M|+i)$th color." In this paper, we first show that the approximation guarantee of the matching based algorithm is $(1 + \frac {2} {\delta})$ for graphs with perfect matching and minimum degree $\delta$. For $\delta \ge 4$, this is better than the $\frac {5} {3}$ approximation guarantee proved in {AAAP}. For triangle free graphs with perfect matching, we prove that the approximation factor is $(1 + \frac {1}{\delta - 1})$, which is better than $5/3$ for $\delta \ge 3$.Comment: 11pages, 3 figure

Sufficient Conditions for Tuza's Conjecture on Packing and Covering Triangles

Given a simple graph $G=(V,E)$, a subset of $E$ is called a triangle cover if it intersects each triangle of $G$. Let $\nu_t(G)$ and $\tau_t(G)$ denote the maximum number of pairwise edge-disjoint triangles in $G$ and the minimum cardinality of a triangle cover of $G$, respectively. Tuza conjectured in 1981 that $\tau_t(G)/\nu_t(G)\le2$ holds for every graph $G$. In this paper, using a hypergraph approach, we design polynomial-time combinatorial algorithms for finding small triangle covers. These algorithms imply new sufficient conditions for Tuza's conjecture on covering and packing triangles. More precisely, suppose that the set $\mathscr T_G$ of triangles covers all edges in $G$. We show that a triangle cover of $G$ with cardinality at most $2\nu_t(G)$ can be found in polynomial time if one of the following conditions is satisfied: (i) $\nu_t(G)/|\mathscr T_G|\ge\frac13$, (ii) $\nu_t(G)/|E|\ge\frac14$, (iii) $|E|/|\mathscr T_G|\ge2$. Keywords: Triangle cover, Triangle packing, Linear 3-uniform hypergraphs, Combinatorial algorithm

Ramsey-nice families of graphs

For a finite family $\mathcal{F}$ of fixed graphs let $R_k(\mathcal{F})$ be the smallest integer $n$ for which every $k$-coloring of the edges of the complete graph $K_n$ yields a monochromatic copy of some $F\in\mathcal{F}$. We say that $\mathcal{F}$ is $k$-nice if for every graph $G$ with $\chi(G)=R_k(\mathcal{F})$ and for every $k$-coloring of $E(G)$ there exists a monochromatic copy of some $F\in\mathcal{F}$. It is easy to see that if $\mathcal{F}$ contains no forest, then it is not $k$-nice for any $k$. It seems plausible to conjecture that a (weak) converse holds, namely, for any finite family of graphs $\mathcal{F}$ that contains at least one forest, and for all $k\geq k_0(\mathcal{F})$ (or at least for infinitely many values of $k$), $\mathcal{F}$ is $k$-nice. We prove several (modest) results in support of this conjecture, showing, in particular, that it holds for each of the three families consisting of two connected graphs with 3 edges each and observing that it holds for any family $\mathcal{F}$ containing a forest with at most 2 edges. We also study some related problems and disprove a conjecture by Aharoni, Charbit and Howard regarding the size of matchings in regular 3-partite 3-uniform hypergraphs.Comment: 20 pages, 2 figure

Vertex decomposable graphs, codismantlability, Cohen-Macaulayness and Castelnuovo-Mumford regularity

We call a (simple) graph G codismantlable if either it has no edges or else it has a codominated vertex x, meaning that the closed neighborhood of x contains that of one of its neighbor, such that G-x codismantlable. We prove that if G is well-covered and it lacks induced cycles of length four, five and seven, than the vertex decomposability, codismantlability and Cohen-Macaulayness for G are all equivalent. The rest deals with the computation of Castelnuovo-Mumford regularity of codismantlable graphs. Note that our approach complements and unifies many of the earlier results on bipartite, chordal and very well-covered graphs