Improved approximation for maximum edge colouring problem

Abstract

The anti-Ramsey number, $ar(G,H)$ is the minimum integer $k$ such that in any edge colouring of $G$ with $k$ colours there is a rainbow subgraph isomorphic to $H$, namely, a copy of $H$ with each of its edges assigned a different colour. The notion was introduced by Erdös and Simonovits in 1973. Since then the parameter has been studied extensively. The case when $H$ is a star graph was considered by several graph theorists from the combinatorial point of view. Recently this case received the attention of researchers from the algorithm community because of its applications in interface modelling of wireless networks. To the algorithm community, the problem is known as maximum edge $q$-colouring problem: Find a colouring of the edges of $G$, maximizing the number of colours satisfying the constraint that each vertex spans at most $q$ colours on its incident edges. It is easy to see that the maximum value of the above optimization problem equals $ar(G,K_{1,q+1})-1$.

In this paper, we study the maximum edge 2-colouring problem from the approximation algorithm point of view. The case $q=2$ is particularly interesting due to its application in real-life problems. Algorithmically, this problem is known to be NP-hard for $q\ge 2$. For the case of $q=2$, it is also known that no polynomial-time algorithm can approximate to a factor less than $3/2$ assuming the unique games conjecture. Feng et al. showed a 2-approximation algorithm for this problem. Later Adamaszek and Popa presented a $5/3$-approximation algorithm with the additional assumption that the input graph has a perfect matching. Note that the obvious but the only known algorithm issues different colours to the edges of a maximum matching (say $M$) and different colours to the connected components of $G\setminus M$. In this article, we give a new analysis of the aforementioned algorithm to show that for triangle-free graphs with perfect matching the approximation ratio is $8/5$. We also show that this algorithm cannot achieve a factor better than $58/37$ on triangle free graphs that has a perfect matching. The contribution of the paper is a completely new, deeper and closer analysis of how the optimum achieves a higher number of colours than the matching based algorithm, mentioned above.

Introduction

A $k$-edge colouring of a graph is a function $f:E\left(G\right)\to \left[k\right]$. Note that $f$ does not need to be a proper colouring of the edges, i.e., edges incident to the same vertex may receive the same colour. A subgraph $H$ of $G$ is called a rainbow subgraph (heterochromatic subgraph) with respect to a $k$-edge colouring $f$ if all the edges of $H$ are coloured distinctly. For a pair of graphs $G$ and $H$ the anti-Ramsey number, $ar(G,H)$, denotes the minimum number of colours $k$ such that in any $k$-edge colouring of $G$ there exists at least one subgraph isomorphic to $H$ which is a rainbow subgraph. Equivalently if $k^{\prime }$ is the maximum possible number of colours in an edge colouring $f$ of $G$ such that there exists no rainbow subgraph isomorphic to $H$ with respect to $f$ then $ar(G,H)=k^{\prime }+1$. We call the first parameter of $ar(G,H)$, $G$ as the input graph and the second parameter $H$ as the pattern graph.

The notion, anti-Ramsey number, was introduced by Erdös and Simonovits in 1973 [6]. Most of the initial research on this topic focused on complete graphs ($K_n$) as the input graph and pattern graphs that possess a certain nice structure, for example, path, cycle, complete graph etc. The exact expression of $ar(K_n,P_k)$, when the pattern graph is a path of length $k$ ($P_k$), was reported in the article written by Simonovits and Sós [22]. On the other hand, the simple case of the pattern graph is a cycle of length $k$ ($C_k$) took years to get solved completely. It was proved by Erdös, Simonovits and Sós that $ar(K_n,C_3)=n-1$ [6]. In the same paper it was conjectured that $ar(K_n,C_n)=(\frac{k-2}{2}+\frac{1}{k-1})n+O\left(1\right)$ for $k\ge 4$. The conjecture was verified affirmatively for the case $k=4$ by Alon [3]. Later it was studied by Jiang and West [14]. Almost thirty years after it was conjectured, Montellano-Ballesteros and Neumann-Lara reported proof of the statement in 2005 [19]. A lower bound considering the pattern graph as the clique of size $n-1$ ($K_{n-1}$) was reported in [16]. Schiermeyer and Montellano-Ballesteros together with Neumann-Lara independently reported the exact value of $ar(K_n,K_r)$ [18], [21]. In the same article Schiermeyer also studied the case when pattern graph is a matching. Haas and Young later studied the case when pattern graph is a perfect matching [11]. A tighter bound on matching was reported in the article by Fujita et al. [8]. The article by Jiang and West reported bounds on $ar(G,H)$ when $H$ is a tree [15]. Jiang also derived an upper bound on $ar(G,H)$ when $H$ is a complete subdivided graph relating the parameter $ar(G,H)$ with Turán number, that is the maximum cardinality of edges of an $n$-vertex graph that does not contain a subgraph $H$ [12].

The study of anti-Ramsey number was not entirely restricted to the case when the input graph is a complete graph. Axenovich et al. studied the case when the input graph is a complete bipartite graph [4]. A $t$-round variant of anti-Ramsey number was introduced and studied in [5].

In this paper, we consider the pattern graph as the claw graph, i.e. the star graph with exactly 3 leaves, denoted by $K_{1,3}$. The study of anti-Ramsey number where the pattern graph is the claw or more generally the star graph was initiated in the work of Manoussakis et al. [16]. The bound was later improved in [13]. In the same article exact value of the bipartite variant of the problem $ar(K_{n,n},K_{1,q})$ was also reported. Gorgol and Lazuka computed the exact value of $ar(G,H)$ when $H$ is $K_{1,4}$ with an edge added to it [9]. Montellano-Ballesteros relaxed the condition on input graph and considered any graph as input in their study [17]. The study of anti-Ramsey number with claw graph as pattern graph was revisited recently due to its application in modelling channel assignment in a network equipped with a multi-channel wireless interface [20]. They introduced the problem as maximum edge $q$-colouring problem, thus initiating the exploration of the algorithmic aspects of this parameter, $ar(G,K_{1,t})$.

For a graph $G$, an edge $q$-colouring of $G$ is an assignment of colours to edges of $G$ such that no more than $q$ distinct colours are incident at any vertex. An optimal edge $q$-colouring is one which uses the maximum number of colours. It is easily seen that the number of colours in maximum edge $q$-colouring of $G$ is $ar(G,K_{1,q+1})-1$.

In [1], it was reported that the problem is $NP$-hard for every $q\ge 2$. Moreover, they showed that it is hard to approximate within a factor of $(1+1/q)$ for every $q\ge 2$, assuming the unique games conjecture. A simple $2$-factor algorithm for maximum $2$-colouring problem was reported in [7]. A description of the algorithm is provided in Algorithm 1.Henceforth we refer to this algorithm as the matching based algorithm. In a recent article [2], authors reported a $5/3$ approximation factor for the same algorithm assuming that the input graphs have a perfect matching. Approximation bounds for the matching based algorithm when the input graph has certain degree constraints were reported in [23]. A fixed-parameter tractable algorithm was reported for the case $q=2$ in [10].

In the present article, our focus is on the case when $q=2$, and when the graph $G$ has a perfect matching. It is worth mentioning here, although Montellano-Ballesteros reported bounds on $ar(G,K_{1,q})$, their expression is not enough to draw any inference in this particular scenario. Their technique is useful for deriving bounds when the input graph has certain regular structures such as complete graph, complete $t$-partite graph, hypercube etc.