Manu Basavaraju — Page 2

arXiv2011

Rainbow Connection Number of Graph Power and Graph Products

Rainbow connection number, rc(G), of a connected graph G is the minimum number of colors needed to color its edges so that every pair of vertices is connected by at least one path in which no two edges are colored the same (Note that the coloring need not be proper). In this paper we study the rainbow connection number with respect to three important graph product operations (namely cartesian product, lexicographic product and strong product) and the operation of taking the power of a graph. In this direction, we show that if G is a graph obtained by applying any of the operations mentioned above on non-trivial graphs, then rc(G) <= 2r(G)+c, where r(G) denotes the radius of G and c \in {0,1,2}. In general the rainbow connection number of a bridgeless graph can be as high as the square of its radius [Basavaraju et. al, 2010]. This is an attempt to identify some graph classes which have rainbow connection number very close to the obvious lower bound of diameter (and thus the radius). The bounds reported are tight upto additive constants. The proofs are constructive and hence yield polynomial time (2 + 2/r(G))-factor approximation algorithms.

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arXiv2010

Acyclic Edge Coloring of Triangle Free Planar Graphs

An $acyclic$ edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The \emph{acyclic chromatic index} of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by $a'(G)$. It was conjectured by Alon, Sudakov and Zaks (and much earlier by Fiamcik) that $a'(G)\le Δ+2$, where $Δ=Δ(G)$ denotes the maximum degree of the graph. If every induced subgraph $H$ of $G$ satisfies the condition $\vert E(H) \vert \le 2\vert V(H) \vert -1$, we say that the graph $G$ satisfies $Property\ A$. In this paper, we prove that if $G$ satisfies $Property\ A$, then $a'(G)\le Δ+ 3$. Triangle free planar graphs satisfy $Property\ A$. We infer that $a'(G)\le Δ+ 3$, if $G$ is a triangle free planar graph. Another class of graph which satisfies $Property\ A$ is 2-fold graphs (union of two forests).

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arXiv2009

Acyclic Edge coloring of Planar Graphs

An $acyclic$ edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The \emph{acyclic chromatic index} of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by $a'(G)$. It was conjectured by Alon, Sudakov and Zaks (and much earlier by Fiamcik) that $a'(G)\le Δ+2$, where $Δ=Δ(G)$ denotes the maximum degree of the graph. We prove that if $G$ is a planar graph with maximum degree $Δ$, then $a'(G)\le Δ+ 12$.

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arXiv2008

Acyclic Edge colorings of 2-degenerate graphs

An $acyclic$ edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycle s. The \emph{acyclic chromatic index} of a graph is the minimum number k such that there is an acyclic e dge coloring using k colors and is denoted by $a'(G)$. A graph is called 2-$degenerate$ if any of its induced subgraph has a vertex of degree at most 2. The class of 2-$degenerate graphs$ properly conta in $series$-$parallel graphs$, $outerplanar graphs$, \emph{non-regular subcubic graphs}, \emph{planar graphs of girth at least 6} and \emph{circle graphs of girth at least 5} as subclasses. It was conjectur ed by Alon, Sudakov and Zaks (and earlier by Fiamcik) that $a'(G)\le Δ+2$, where $Δ=Δ(G)$ denotes the maximum deg ree of the graph. We prove the conjecture for 2-$degenerate$ graphs: in fact we prove a stronger bound . We prove that if $G$ is a 2-degenerate graph with maximum degree $Δ$, then $a'(G)\le Δ+ 1$.

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arXiv2008

$d$-Regular graphs of acyclic chromatic index at least $d+2$

An $acyclic$ edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The \emph{acyclic chromatic index} of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by $a'(G)$. It was conjectured by Alon, Sudakov and Zaks (and earlier by Fiamcik) that $a'(G)\le Δ+2$, where $Δ=Δ(G)$ denotes the maximum degree of the graph. Alon et.al also raised the question whether the complete graphs of even order are the only regular graphs which require $Δ+2$ colors to be acyclically edge colored. In this paper, using a simple counting argument we observe not only that this is not true, but infact all d-regular graphs with $2n$ vertices and $d > n$, requires at least $d+2$ colors. We also show that $a'(K_{n,n}) \ge n+2$, when $n$ is odd using a more non-trivial argument(Here $K_{n,n}$ denotes the complete bipartite graph with $n$ vertices on each side). This lower bound for $K_{n,n}$ can be shown to be tight for some families of complete bipartite graphs and for small values of $n$. We also infer that for every $d,n$ such that $d \ge 5$, $n \ge 2d + 3$ and $dn$ even, there exist $d$-regular graphs which require at least $d+2$-colors to be acyclically edge colored.

math.CO↗

arXiv2008

Acyclic Edge Coloring of Graphs with Maximum Degree 4

An $acyclic$ edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycle s. The \emph{acyclic chromatic index} of a graph is the minimum number k such that there is an acyclic e dge coloring using k colors and is denoted by $a'(G)$. It was conjectured by Alon, Sudakov and Zaks that for any simple and finite graph $G$, $a'(G)\le Δ+2$, where $Δ=Δ(G)$ denotes the maximum degree of $G$. We prove the conjecture for connected graphs with $Δ(G) \le 4$, with the additional restriction that $m \le 2n-1$, where $n$ is the number of vertices and $m$ is the number of edges in $G $. Note that for any graph $G$, $m \le 2n$, when $Δ(G) \le 4$. It follows that for any graph $G$ if $Δ(G) \le 4$, then $a'(G) \le 7$.

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