Showing papers by "Deepak Rajendraprasad published in 2015"

Separation Dimension of Bounded Degree Graphs

Abstract: The separation dimension of a graph $G$ is the smallest natural number $k$ for which the vertices of $G$ can be embedded in $\mathbb{R}^k$ such that any pair of disjoint edges in $G$ can be separated by a hyperplane normal to one of the axes. Equivalently, it is the smallest possible cardinality of a family $\mathcal{F}$ of total orders of the vertices of $G$ such that for any two disjoint edges of $G$, there exists at least one total order in $\mathcal{F}$ in which all the vertices in one edge precede those in the other. In general, the maximum separation dimension of a graph on $n$ vertices is $\Theta(\log n)$. In this article, we focus on bounded degree graphs and show that the separation dimension of a graph with maximum degree $d$ is at most $2^{9{log^{\star}}\!d} d$. We also demonstrate that the above bound is nearly tight by showing that, for every $d$, almost all $d$-regular graphs have separation dimension at least $\ceil{d/2}$.

Boxicity and cubicity of product graphs

Abstract: The boxicity (cubicity) of a graph G is the minimum natural number k such that G can be represented as an intersection graph of axis-parallel rectangular boxes (axis-parallel unit cubes) in R k . In this article, we give estimates on the boxicity and the cubicity of Cartesian, strong and direct products of graphs in terms of invariants of the component graphs. In particular, we study the growth, as a function of d , of the boxicity and the cubicity of the d th power of a graph with respect to the three products. Among others, we show a surprising result that the boxicity and the cubicity of the d th Cartesian power of any given finite graph is, respectively, in O ( log d / log log d ) and ? ( d / log d ) . On the other hand, we show that there cannot exist any sublinear bound on the growth of the boxicity of powers of a general graph with respect to strong and direct products.

Heterochromatic paths in edge colored graphs without small cycles and heterochromatic-triangle-free graphs

Abstract: Conditions for the existence of heterochromatic Hamiltonian paths and cycles in edge colored graphs are well investigated in literature. A related problem in this domain is to obtain good lower bounds for the length of a maximum heterochromatic path in an edge colored graph G . This problem is also well explored by now and the lower bounds are often specified as functions of the minimum color degree of G -the minimum number of distinct colors occurring at edges incident to any vertex of G -denoted by ? ( G ) .Initially, it was conjectured that the lower bound for the length of a maximum heterochromatic path for an edge colored graph G would be ? 2 ? ( G ) 3 ? . Chen and Li (2005) showed that the length of a maximum heterochromatic path in an edge colored graph G is at least ? ( G ) - 1 , if 1 ? ? ( G ) ? 7 , and at least ? 3 ? ( G ) 5 ? + 1 , if ? ( G ) ? 8 . They conjectured that the tight lower bound would be ? ( G ) - 1 and demonstrated some examples which achieve this bound. An unpublished manuscript from the same authors (Chen, Li) reported to show that if ? ( G ) ? 8 , then G contains a heterochromatic path of length at least ? 2 ? ( G ) 3 ? + 1 .In this paper, we give lower bounds for the length of a maximum heterochromatic path in edge colored graphs without small cycles. We show that if G has no four cycles, then it contains a heterochromatic path of length at least ? ( G ) - o ( ? ( G ) ) and if the girth of G is at least 4 log 2 ( ? ( G ) ) + 2 , then it contains a heterochromatic path of length at least ? ( G ) - 2 , which is only one less than the bound conjectured by Chen and Li (2005). Other special cases considered include lower bounds for the length of a maximum heterochromatic path in edge colored bipartite graphs and triangle-free graphs: for triangle-free graphs we obtain a lower bound of ? 5 ? ( G ) 6 ? and for bipartite graphs we obtain a lower bound of ? 6 ? ( G ) - 3 7 ? .In this paper, it is also shown that if the coloring is such that G has no heterochromatic triangles, then G contains a heterochromatic path of length at least ? 13 ? ( G ) 17 ? . This improves the previously known ? 3 ? ( G ) 4 ? bound obtained by Chen and Li (2011). We also give a relatively shorter and simpler proof showing that any edge colored graph G contains a heterochromatic path of length at least ? 2 ? ( G ) 3 ? .

Boundaries of Hypertrees, and Hamiltonian Cycles in Simplicial Complexes

Abstract: A $d$-hypertree on $[n]$ is a maximal acyclic $d$-dimensional simplicial complex with full $(d-1)$-skeleton on the vertex set $[n]$. Alternatively, in the language of algebraic topology, it is a minimal $d$-dimensional simplicial complex $T$ (assuming full $(d-1)$-skeleton) such that $\tilde{H}_{d-1}(T;\mathbb{F})=0$. The $d$-hypertrees are a basic object in combinatorial theory of simplicial complexes. They have been studied; and yet, many of their structural aspects remain poorly understood. In this paper we study the boundaries $\partial_d T$ of $d$-hypertrees, and the fundamental $d$-cycles defined by them. Our findings include: 1. A full characterization of $\partial_d T$ over $\mathbb{F}_2$ for $d \leq 2$, and some partial results for $d \geq 3$. 2. Lower bounds on the maximum size of a largest simple $d$-cycle on $[n]$. In particular, for $d=2$, we construct a {\em Hamiltonian $d$-cycle} $H$ on $[n]$, i.e., a simple $d$-cycle of size ${{n-1} \choose d} + 1$. For $d\geq 3$, we construct a simple $d$-cycle of size ${{n-1} \choose d} - O(n^{d-2})$. 3. Observing that the maximum of the expected distance between two vertices chosen uniformly at random in a tree ($1$-hypertree) on $[n]$ is at most $\thicksim n/3$, attained on Hamiltonian paths, we ask a similar question about $d$-hypertrees. "How large can be the {\em average} size of a fundamental cycle of a $d$-hypertree $T$ (i.e., the expected size of the dependency created by adding a $d$-simplex on $[n]$, chosen uniformly at random, to $T$)?" For every $d \in \mathbb{N}$, we construct an infinite family of $d$-hypertrees $\{T\}$ with the average size of a fundamental cycle at least $c_d\, |T| \,=\, c_d\,{n-1 \choose d}$, where $c_d$ is a constant depending on the dimension $d$ alone.

Partial List Colouring of Certain Graphs

Abstract: The partial list colouring conjecture due to Albertson, Grossman, and Haas (2000) states that for every $s$-choosable graph $G$ and every assignment of lists of size $t$, $1 \leq t \leq s$, to the vertices of $G$ there is an induced subgraph of $G$ on at least $\frac{t|V(G)|}{s}$ vertices which can be properly coloured from these lists. In this paper, we show that the partial list colouring conjecture holds true for certain classes of graphs like claw-free graphs, graphs with chromatic number at least $\frac{|V(G)|-1}{2}$, chordless graphs, and series-parallel graphs.