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Register a new userApproximation bounds on maximum edge 2-coloring of dense graphs
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Added by
L. Sunil Chandran
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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For a graph $G$ and integer $qgeq 2$, 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 $qgeq 2$ 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 $delta ge 3$.
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An incidence of an undirected graph G is a pair $(v,e)$ where $v$ is a vertex of $G$ and $e$ an edge of $G$ incident with $v$. Two incidences $(v,e)$ and $(w,f)$ are adjacent if one of the following holds: (i) $v = w$, (ii) $e = f$ or (iii) $vw = e$ or $f$. An incidence coloring of $G$ assigns a color to each incidence of $G$ in such a way that adjacent incidences get distinct colors. In 2005, Hosseini Dolama emph{et al.}~citep{ds05} proved that every graph with maximum average degree strictly less than $3$ can be incidence colored with $Delta+3$ colors. Recently, Bonamy emph{et al.}~citep{Bonamy} proved that every graph with maximum degree at least $4$ and with maximum average degree strictly less than $frac{7}{3}$ admits an incidence $(Delta+1)$-coloring. In this paper we give bounds for the number of colors needed to color graphs having maximum average degrees bounded by different values between $4$ and $6$. In particular we prove that every graph with maximum degree at least $7$ and with maximum average degree less than $4$ admits an incidence $(Delta+3)$-coloring. This result implies that every triangle-free planar graph with maximum degree at least $7$ is incidence $(Delta+3)$-colorable. We also prove that every graph with maximum average degree less than 6 admits an incidence $(Delta + 7)$-coloring. More generally, we prove that $Delta+k-1$ colors are enough when the maximum average degree is less than $k$ and the maximum degree is sufficiently large.
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 $H$, i.e., a copy of $H$ with each of its edges assigned a different colour. The notion was introduced by Erd{{o}}s and Simonovits in 1973. Since then the parameter has been studied extensively in combinatorics, also the particular case when $H$ is a star graph. 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. 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 $qge 2$. 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 leading to an improved approximation bound for triangle-free graphs with perfect matching. We also show a new lower bound when the input graph is triangle-free. 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.
An edge-coloring of a graph $G$ with colors $1,2,ldots,t$ is an interval $t$-coloring if all colors are used, and the colors of edges incident to each vertex of $G$ are distinct and form an interval of integers. A graph $G$ is interval colorable if it has an interval $t$-coloring for some positive integer $t$. For an interval colorable graph $G$, $W(G)$ denotes the greatest value of $t$ for which $G$ has an interval $t$-coloring. It is known that the complete graph is interval colorable if and only if the number of its vertices is even. However, the exact value of $W(K_{2n})$ is known only for $n leq 4$. The second author showed that if $n = p2^q$, where $p$ is odd and $q$ is nonnegative, then $W(K_{2n}) geq 4n-2-p-q$. Later, he conjectured that if $n in mathbb{N}$, then $W(K_{2n}) = 4n - 2 - leftlfloorlog_2{n}rightrfloor - left | n_2 right |$, where $left | n_2 right |$ is the number of $1$s in the binary representation of $n$. In this paper we introduce a new technique to construct interval colorings of complete graphs based on their 1-factorizations, which is used to disprove the conjecture, improve lower and upper bounds on $W(K_{2n})$ and determine its exact values for $n leq 12$.
This note resolves an open problem asked by Bezrukov in the open problem session of IWOCA 2014. It shows an equivalence between regular graphs and graphs for which a sequence of invariants presents some symmetric property. We extend this result to a few other sequences.
Clique-width is a complexity measure of directed as well as undirected graphs. Rank-width is an equivalent complexity measure for undirected graphs and has good algorithmic and structural properties. It is in particular related to the vertex-minor relation. We discuss an extension of the notion of rank-width to edge-colored graphs. A C-colored graph is a graph where the arcs are colored with colors from the set C. There is not a natural notion of rank-width for C-colored graphs. We define two notions of rank-width for them, both based on a coding of C-colored graphs by edge-colored graphs where each edge has exactly one color from a field F and named respectively F-rank-width and F-bi-rank-width. The two notions are equivalent to clique-width. We then present a notion of vertex-minor for F-colored graphs and prove that F-colored graphs of bounded F-rank-width are characterised by a finite list of F-colored graphs to exclude as vertex-minors. A cubic-time algorithm to decide whether a F-colored graph has F-rank-width (resp. F-bi-rank-width) at most k, for fixed k, is also given. Graph operations to check MSOL-definable properties on F-colored graphs of bounded rank-width are presented. A specialisation of all these notions to (directed) graphs without edge colors is presented, which shows that our results generalise the ones in undirected graphs.
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