Search

A graph $G$ on $n$ vertices is a \emph{threshold graph} if there exist real numbers $a_1,a_2, \ldots, a_n$ and $b$ such that the zero-one solutions of the linear inequality $\sum \limits_{i=1}^n a_i x_i \leq b$ are the characteristic vectors of the cliques of $G$. Introduced in [Chv{\'a}tal and Hammer, Annals of Discrete Mathematics, 1977], the \emph{threshold dimension} of a graph $G$, denoted by $\dimth(G)$, is the minimum number of threshold graphs whose intersection yields $G$. Given a graph $G$ on $n$ vertices, in line with Chv{\'a}tal and Hammer, $f_G\colon \{0,1\}^n \rightarrow \{0,1\}$ is the Boolean function that has the property that $f_G(x) = 1$ if and only if $x$ is the characteristic vector of a clique in $G$. A Boolean function $f$ for which there exists a graph $G$ such that $f=f_G$ is called a \emph{graphic} Boolean function. It follows that for a graph $G$, $\dimth(G)$ is precisely the minimum number of \emph{majority} gates whose AND (or conjunction) realizes the graphic Boolean function $f_G$. The fact that there exist Boolean functions which can be realized as the AND of only exponentially many majority gates motivates us to study threshold dimension of graphs. We give tight or nearly tight upper bounds for the threshold dimension of a graph in terms of its treewidth, maximum degree, degeneracy, number of vertices, size of a minimum vertex cover, etc. We also study threshold dimension of random graphs and graphs with high girth.

Given a finite family $\mathcal{F}$ of graphs, we say that a graph $G$ is "$\mathcal{F}$-free" if $G$ does not contain any graph in $\mathcal{F}$ as a subgraph. A vertex-colored graph $H$ is called "rainbow" if no two vertices of $H$ have the same color. Given an integer $s$ and a finite family of graphs $\mathcal{F}$, let $\ell(s,\mathcal{F})$ denote the smallest integer such that any properly vertex-colored $\mathcal{F}$-free graph $G$ having $\chi(G)\geq\ell(s,\mathcal{F})$ contains an induced rainbow path on $s$ vertices. Scott and Seymour showed that $\ell(s,K)$ exists for every complete graph $K$. A conjecture of N. R. Aravind states that $\ell(s,C_3)=s$. The upper bound on $\ell(s,C_3)$ that can be obtained using the methods of Scott and Seymour setting $K=C_3$ are, however, super-exponential. Gy\'arf\'as and S\'ark\"ozy showed that $\ell(s,\{C_3,C_4\})=\mathcal{O}\big((2s)^{2s}\big)$. For $r\geq 2$, we show that $\ell(s,K_{2,r})\leq (r-1)(s-1)(s-2)/2+s$ and therefore, $\ell(s,C_4)\leq\frac{s^2-s+2}{2}$. This significantly improves Gy\'arf\'as and S\'ark\"ozy's bound and also covers a bigger class of graphs. We adapt our proof to achieve much stronger upper bounds for graphs of higher girth: we prove that $\ell(s,\{C_3,C_4,\ldots,C_{g-1}\})\leq s^{1+\frac{4}{g-4}}$, where $g\geq 5$. Moreover, in each case, our results imply the existence of at least $s!/2$ distinct induced rainbow paths on $s$ vertices. Along the way, we obtain some results on related problems on oriented graphs. For $r\geq 2$, let $\mathcal{B}_r$ denote the orientations of $K_{2,r}$ in which one vertex has out-degree or in-degree $r$. We show that every $\mathcal{B}_r$-free oriented graph $G$ having $\chi(G)\geq (r-1)(s-1)(s-2)+2s+1$ and every bikernel-perfect oriented graph $G$ with girth $g\geq 5$ having $\chi(G)\geq 2s^{1+\frac{4}{g-4}}$ contains every $s$ vertex oriented tree as an induced subgraph., Comment: 21 pages, 1 figure

Hadwiger's conjecture is one of the most important and long-standing conjectures in graph theory. Reed and Seymour showed in 2004 that Hadwiger's conjecture is true for line graphs. We investigate this conjecture on the closely related class of total graphs. The total graph of $G$, denoted by $T(G)$, is defined on the vertex set $V(G)\sqcup E(G)$ with $c_1,c_2\in V(G)\sqcup E(G)$ adjacent whenever $c_1$ and $c_2$ are adjacent to or incident on each other in $G$. We first show that there exists a constant $C$ such that, if the connectivity of $G$ is at least $C$, then Hadwiger's conjecture is true for $T(G)$. The total chromatic number $\chi"(G)$ of a graph $G$ is defined to be equal to the chromatic number of its total graph. That is, $\chi"(G)=\chi(T(G))$. Another well-known conjecture in graph theory, the total coloring conjecture or TCC, states that for every graph $G$, $\chi"(G)\leq\Delta(G)+2$, where $\Delta(G)$ is the maximum degree of $G$. We show that if a weaker version of the total coloring conjecture (weak TCC) namely, $\chi"(G)\leq\Delta(G)+3$, is true for a class of graphs $\mathcal{F}$ that is closed under the operation of taking subgraphs, then Hadwiger's conjecture is true for the class of total graphs of graphs in $\mathcal{F}$. This motivated us to look for classes of graphs that satisfy weak TCC. It may be noted that a complete proof of TCC for even 4-colorable graphs (in fact even for planar graphs) has remained elusive even after decades of effort; but weak TCC can be proved easily for 4-colorable graphs. We noticed that in spite of the interest in studying $\chi"(G)$ in terms of $\chi(G)$ right from the initial days, weak TCC is not proven to be true for $k$-colorable graphs even for $k=5$. In the second half of the paper, we make a contribution to the literature on total coloring by proving that $\chi"(G)\leq\Delta(G)+3$ for every 5-colorable graph $G$.

Given a digraph $G$, a set $X\subseteq V(G)$ is said to be absorbing set (resp. dominating set) if every vertex in the graph is either in $X$ or is an in-neighbour (resp. out-neighbour) of a vertex in $X$. A set $S\subseteq V(G)$ is said to be an independent set if no two vertices in $S$ are adjacent in $G$. A kernel (resp. solution) of $G$ is an independent and absorbing (resp. dominating) set in $G$. We explore the algorithmic complexity of these problems in the well known class of interval digraphs. A digraph $G$ is an interval digraph if a pair of intervals $(S_u,T_u)$ can be assigned to each vertex $u$ of $G$ such that $(u,v)\in E(G)$ if and only if $S_u\cap T_v\neq\emptyset$. Many different subclasses of interval digraphs have been defined and studied in the literature by restricting the kinds of pairs of intervals that can be assigned to the vertices. We observe that several of these classes, like interval catch digraphs, interval nest digraphs, adjusted interval digraphs and chronological interval digraphs, are subclasses of the more general class of reflexive interval digraphs -- which arise when we require that the two intervals assigned to a vertex have to intersect. We show that all the problems mentioned above are efficiently solvable, in most of the cases even linear-time solvable, in the class of reflexive interval digraphs, but are APX-hard on even the very restricted class of interval digraphs called point-point digraphs, where the two intervals assigned to each vertex are required to be degenerate, i.e. they consist of a single point each. The results we obtain improve and generalize several existing algorithms and structural results for subclasses of reflexive interval digraphs., Comment: 26 pages, 3 figures

A graph $G$ is said to be the intersection of graphs $G_1,G_2,\ldots,G_k$ if $V(G)=V(G_1)=V(G_2)=\cdots=V(G_k)$ and $E(G)=E(G_1)\cap E(G_2)\cap\cdots\cap E(G_k)$. For a graph $G$, $\mathrm{dim}_{COG}(G)$ (resp. $\mathrm{dim}_{TH}(G)$) denotes the minimum number of cographs (resp. threshold graphs) whose intersection gives $G$. We present several new bounds on these parameters for general graphs as well as some special classes of graphs. It is shown that for any graph $G$: (a) $\mathrm{dim}_{COG}(G)\leq\mathrm{tw}(G)+2$, (b) $\mathrm{dim}_{TH}(G)\leq\mathrm{pw}(G)+1$, and (c) $\mathrm{dim}_{TH}(G)\leq\chi(G)\cdot\mathrm{box}(G)$, where $\mathrm{tw}(G)$, $\mathrm{pw}(G)$, $\chi(G)$ and $\mathrm{box}(G)$ denote respectively the treewidth, pathwidth, chromatic number and boxicity of the graph $G$. We also derive the exact values for these parameters for cycles and show that every forest is the intersection of two cographs. These results allow us to derive improved bounds on $\mathrm{dim}_{COG}(G)$ and $\mathrm{dim}_{TH}(G)$ when $G$ belongs to some special graph classes., Comment: 14 pages, 4 figures, accepted for the conference CALDAM 2020

Threshold graphs are a class of graphs that have many equivalent definitions and have applications in integer programming and set packing problems. A graph is said to have a threshold cover of size $k$ if its edges can be covered using $k$ threshold graphs. Chv\'atal and Hammer, in 1977, defined the threshold dimension $\mathrm{th}(G)$ of a graph $G$ to be the least integer $k$ such that $G$ has a threshold cover of size $k$ and observed that $\mathrm{th}(G)\geq\chi(G^*)$, where $G^*$ is a suitably constructed auxiliary graph. Raschle and Simon~[Proceedings of the Twenty-seventh Annual ACM Symposium on Theory of Computing, STOC '95, pages 650--661, 1995] proved that $\mathrm{th}(G)=\chi(G^*)$ whenever $G^*$ is bipartite. We show how the lexicographic method of Hell and Huang can be used to obtain a completely new and, we believe, simpler proof for this result. For the case when $G$ is a split graph, our method yields a proof that is much shorter than the ones known in the literature., Comment: 14 pages

Planar graphs are the graphs with Dushnik-Miller dimension at most three (W. Schnyder, Planar graphs and poset dimension, Order 5, 323-343, 1989). Consider the intersection graph of interior disjoint axis parallel rectangles in the plane. It is known that if at most three rectangles intersect on a point, then this intersection graph is planar, that is it has Dushnik-Miller dimension at most three. This paper aims at generalizing this from the plane to $R^d$ by considering tilings of $R^d$ with axis parallel boxes, where at most $d+1$ boxes intersect on a point. Such tilings induce simplicial complexes and we will show that those simplicial complexes have Dushnik-Miller dimension at most $d+1$., Comment: 14 pages

A vertex in a directed graph is said to have a large second neighborhood if it has at least as many second out-neighbors as out-neighbors. The Second Neighborhood Conjecture, first stated by Seymour, asserts that there is a vertex having a large second neighborhood in every oriented graph (a directed graph without loops or digons). We prove that oriented graphs whose missing edges can be partitioned into a (possibly empty) matching and a (possibly empty) star satisfy this conjecture. This generalizes a result of Fidler and Yuster. An implication of our result is that every oriented graph without a sink and whose missing edges form a (possibly empty) matching has at least two vertices with large second neighborhoods. This is a strengthening of a theorem of Havet and Thomasse, who showed that the same holds for tournaments without a sink. Moreover, we also show that the conjecture is true for oriented graphs whose vertex set can be partitioned into an independent set and a 2-degenerate graph., Comment: 23 pages, 2 figures

We introduce the notion of \emph{stab number} and \emph{exact stab number} of rectangle intersection graphs, otherwise known as graphs of boxicity at most 2. A graph $G$ is said to be a \emph{$k$-stabbable rectangle intersection graph}, or \emph{$k$-SRIG} for short, if it has a rectangle intersection representation in which $k$ horizontal lines can be chosen such that each rectangle is intersected by at least one of them. If there exists such a representation with the additional property that each rectangle intersects exactly one of the $k$ horizontal lines, then the graph $G$ is said to be a \emph{$k$-exactly stabbable rectangle intersection graph}, or \emph{$k$-ESRIG} for short. The stab number of a graph $G$, denoted by $stab(G)$, is the minimum integer $k$ such that $G$ is a $k$-SRIG. Similarly, the exact stab number of a graph $G$, denoted by $estab(G)$, is the minimum integer $k$ such that $G$ is a $k$-ESRIG. In this work, we study the stab number and exact stab number of some subclasses of rectangle intersection graphs. A lower bound on the stab number of rectangle intersection graphs in terms of its pathwidth and clique number is shown. Tight upper bounds on the exact stab number of split graphs with boxicity at most 2 and block graphs are also given. We show that for $k\leq 3$, $k$-SRIG is equivalent to $k$-ESRIG and for any $k\geq 10$, there is a tree which is a $k$-SRIG but not a $k$-ESRIG. We also develop a forbidden structure characterization for block graphs that are 2-ESRIG and trees that are 3-ESRIG, which lead to polynomial-time recognition algorithms for these two classes of graphs. These forbidden structures are natural generalizations of asteroidal triples. Finally, we construct examples to show that these forbidden structures are not sufficient to characterize block graphs that are 3-SRIG or trees that are $k$-SRIG for any $k\geq 4$.

A matching $M$ in a graph $G$ is said to be uniquely restricted if there is no other matching in $G$ that matches the same set of vertices as $M$. We describe a polynomial-time algorithm to compute a maximum cardinality uniquely restricted matching in an interval graph, thereby answering a question of Golumbic et al. ("Uniquely restricted matchings", M. C. Golumbic, T. Hirst and M. Lewenstein, Algorithmica, 31:139--154, 2001). Our algorithm actually solves the more general problem of computing a maximum cardinality "strong independent set" in an interval nest digraph, which may be of independent interest. Further, we give linear-time algorithms for computing maximum cardinality uniquely restricted matchings in proper interval graphs and bipartite permutation graphs., Comment: 18 pages, 3 figures

Given a graph $G=(V,E)$ whose vertices have been properly coloured, we say that a path in $G$ is "colourful" if no two vertices in the path have the same colour. It is a corollary of the Gallai-Roy-Vitaver Theorem that every properly coloured graph contains a colourful path on $\chi(G)$ vertices. We explore a conjecture that states that every properly coloured triangle-free graph $G$ contains an induced colourful path on $\chi(G)$ vertices and prove its correctness when the girth of $G$ is at least $\chi(G)$. Recent work on this conjecture by Gy\'arf\'as and S\'ark\"ozy, and Scott and Seymour has shown the existence of a function $f$ such that if $\chi(G)\geq f(k)$, then an induced colourful path on $k$ vertices is guaranteed to exist in any properly coloured triangle-free graph $G$., Comment: 8 pages, 3 figures

A piecewise linear curve in the plane made up of $k+1$ line segments, each of which is either horizontal or vertical, with consecutive segments being of different orientation is called a $k$-bend path. Given a graph $G$, a collection of $k$-bend paths in which each path corresponds to a vertex in $G$ and two paths have a common point if and only if the vertices corresponding to them are adjacent in $G$ is called a $B_k$-VPG representation of $G$. Similarly, a collection of $k$-bend paths each of which corresponds to a vertex in $G$ is called an $B_k$-EPG representation of $G$ if any two paths have a line segment of non-zero length in common if and only if their corresponding vertices are adjacent in $G$. The VPG bend-number $b_v(G)$ of a graph $G$ is the minimum $k$ such that $G$ has a $B_k$-VPG representation. Similarly, the EPG bend-number $b_e(G)$ of a graph $G$ is the minimum $k$ such that $G$ has a $B_k$-EPG representation. Halin graphs are the graphs formed by taking a tree with no degree $2$ vertex and then connecting its leaves to form a cycle in such a way that the graph has a planar embedding. We prove that if $G$ is a Halin graph then $b_v(G) \leq 1$ and $b_e(G) \leq 2$. These bounds are tight. In fact, we prove the stronger result that if $G$ is a planar graph formed by connecting the leaves of any tree to form a simple cycle, then it has a VPG-representation using only one type of 1-bend paths and an EPG-representation using only one type of 2-bend paths., Comment: 11 pages, 3 figures

A strong edge colouring of a graph is an assignment of colours to the edges of the graph such that for every colour, the set of edges that are given that colour form an induced matching in the graph. The strong chromatic index of a graph $G$, denoted by $\chi'_s(G)$, is the minimum number of colours needed in any strong edge colouring of $G$. A graph is said to be \emph{chordless} if there is no cycle in the graph that has a chord. Faudree, Gy\'arf\'as, Schelp and Tuza~[The Strong Chromatic Index of Graphs, Ars Combin., 29B (1990), pp.~205--211] considered a particular subclass of chordless graphs, namely the class of graphs in which all the cycle lengths are multiples of four, and asked whether the strong chromatic index of these graphs can be bounded by a linear function of the maximum degree. Chang and Narayanan~[Strong Chromatic Index of 2-degenerate Graphs, J. Graph Theory, 73(2) (2013), pp.~119--126] answered this question in the affirmative by proving that if $G$ is a chordless graph with maximum degree $\Delta$, then $\chi'_s(G) \leq 8\Delta -6$. We improve this result by showing that for every chordless graph $G$ with maximum degree $\Delta$, $\chi'_s(G)\leq 3\Delta$. This bound is tight up to to an additive constant., Comment: 8 pages + 2 page appendix, 1 figure

Multiple interval graphs are variants of interval graphs where instead of a single interval, each vertex is assigned a set of intervals on the real line. We study the complexity of the MAXIMUM CLIQUE problem in several classes of multiple interval graphs. The MAXIMUM CLIQUE problem, or the problem of finding the size of the maximum clique, is known to be NP-complete for $t$-interval graphs when $t\geq 3$ and polynomial-time solvable when $t=1$. The problem is also known to be NP-complete in $t$-track graphs when $t\geq 4$ and polynomial-time solvable when $t\leq 2$. We show that MAXIMUM CLIQUE is already NP-complete for unit 2-interval graphs and unit 3-track graphs. Further, we show that the problem is APX-complete for 2-interval graphs, 3-track graphs, unit 3-interval graphs and unit 4-track graphs. We also introduce two new classes of graphs called $t$-circular interval graphs and $t$-circular track graphs and study the complexity of the MAXIMUM CLIQUE problem in them. On the positive side, we present a polynomial time $t$-approximation algorithm for WEIGHTED MAXIMUM CLIQUE on $t$-interval graphs, improving earlier work with approximation ratio $4t$., Comment: 22 pages

A graph is said to be a segment graph if its vertices can be mapped to line segments in the plane such that two vertices have an edge between them if and only if their corresponding line segments intersect. Kratochv\'{i}l and Kub\v{e}na [``On intersection representations of co-planar graphs'', Discrete Mathematics, 178(1-3):251-255, 1998] asked the question of whether the complements of planar graphs are segment graphs. We show here that the complements of all partial 2-trees are segment graphs., Comment: 5 pages, 4 figures, preliminary version

The boxicity of a graph G is defined as the minimum integer k such that G is an intersection graph of axis-parallel k-dimensional boxes. Chordal bipartite graphs are bipartite graphs that do not contain an induced cycle of length greater than 4. It was conjectured by Otachi, Okamoto and Yamazaki that chordal bipartite graphs have boxicity at most 2. We disprove this conjecture by exhibiting an infinite family of chordal bipartite graphs that have unbounded boxicity., Comment: 9 pages, 1 figure

The boxicity of a graph G, denoted as box(G) is defined as the minimum integer t such that G is an intersection graph of axis-parallel t-dimensional boxes. A graph G is a k-leaf power if there exists a tree T such that the leaves of the tree correspond to the vertices of G and two vertices in G are adjacent if and only if their corresponding leaves in T are at a distance of at most k. Leaf powers are a subclass of strongly chordal graphs and are used in the construction of phylogenetic trees in evolutionary biology. We show that for a k-leaf power G, box(G)\leq k-1. We also show the tightness of this bound by constructing a k-leaf power with boxicity equal to k-1. This result implies that there exists strongly chordal graphs with arbitrarily high boxicity which is somewhat counterintuitive.

A unit cube in $k$ dimensions ($k$-cube) is defined as the the Cartesian product $R_1\times R_2\times...\times R_k$ where $R_i$(for $1\leq i\leq k$) is a closed interval of the form $[a_i,a_i+1]$ on the real line. A graph $G$ on $n$ nodes is said to be representable as the intersection of $k$-cubes (cube representation in $k$ dimensions) if each vertex of $G$ can be mapped to a $k$-cube such that two vertices are adjacent in $G$ if and only if their corresponding $k$-cubes have a non-empty intersection. The \emph{cubicity} of $G$ denoted as $\cubi(G)$ is the minimum $k$ for which $G$ can be represented as the intersection of $k$-cubes. We give an $O(bw\cdot n)$ algorithm to compute the cube representation of a general graph $G$ in $bw+1$ dimensions given a bandwidth ordering of the vertices of $G$, where $bw$ is the \emph{bandwidth} of $G$. As a consequence, we get $O(\Delta)$ upper bounds on the cubicity of many well-known graph classes such as AT-free graphs, circular-arc graphs and co-comparability graphs which have $O(\Delta)$ bandwidth. Thus we have: 1) $\cubi(G)\leq 3\Delta-1$, if $G$ is an AT-free graph. 2) $\cubi(G)\leq 2\Delta+1$, if $G$ is a circular-arc graph. 3) $\cubi(G)\leq 2\Delta$, if $G$ is a co-comparability graph. Also for these graph classes, there are constant factor approximation algorithms for bandwidth computation that generate orderings of vertices with $O(\Delta)$ width. We can thus generate the cube representation of such graphs in $O(\Delta)$ dimensions in polynomial time., Comment: 9 pages, 0 figures

A k-dimensional box is the Cartesian product R_1 x R_2 x ... x R_k where each R_i is a closed interval on the real line. The boxicity of a graph G, denoted as box(G) is the minimum integer k such that G is the intersection graph of a collection of k-dimensional boxes. Halin graphs are the graphs formed by taking a tree with no degree 2 vertex and then connecting its leaves to form a cycle in such a way that the graph has a planar embedding. We prove that if G is a Halin graph that is not isomorphic to K_4, then box(G)=2. In fact, we prove the stronger result that if G is a planar graph formed by connecting the leaves of any tree in a simple cycle, then box(G)=2 unless G is isomorphic to K_4 (in which case its boxicity is 1)., Comment: 9 pages

An axis-parallel $d$--dimensional box is a Cartesian product $R_1 \times R_2 \times ... \times R_d$ where $R_i$ (for $1 \le i \le d$) is a closed interval of the form $[a_i, b_i]$ on the real line. For a graph $G$, its \emph{boxicity} $\boxi(G)$ is the minimum dimension $d$, such that $G$ is representable as the intersection graph of (axis--parallel) boxes in $d$--dimensional space. The concept of boxicity finds applications in various areas such as ecology, operation research etc. We show that for any graph $G$ with maximum degree $\Delta$, $\boxi(G) \le 2 \Delta^2 + 2$. That the bound does not depend on the number of vertices is a bit surprising considering the fact that there are highly connected bounded degree graphs such as expander graphs. Our proof is very short and constructive. We conjecture that $\boxi(G)$ is $O(\Delta)$., Comment: 4 pages

A unit cube in $k$ dimensional space (or \emph{$k$-cube} in short) is defined as the Cartesian product $R_1\times R_2\times...\times R_k$ where $R_i$(for $1\leq i\leq k$) is a closed interval of the form $[a_i,a_i+1]$ on the real line. A $k$-cube representation of a graph $G$ is a mapping of the vertices of $G$ to $k$-cubes such that two vertices in $G$ are adjacent if and only if their corresponding $k$-cubes have a non-empty intersection. The \emph{cubicity} of $G$, denoted as $\cubi(G)$, is the minimum $k$ such that $G$ has a $k$-cube representation. Roberts \cite{Roberts} showed that for any graph $G$ on $n$ vertices, $\cubi(G)\leq 2n/3$. Many NP-complete graph problems have polynomial time deterministic algorithms or have good approximation ratios in graphs of low cubicity. In most of these algorithms, computing a low dimensional cube representation of the given graph is usually the first step. We present an efficient algorithm to compute the $k$-cube representation of $G$ with maximum degree $\Delta$ in $O(\Delta \ln b)$ dimensions where $b$ is the bandwidth of $G$. Bandwidth of $G$ is at most $n$ and can be much lower. The algorithm takes as input a bandwidth ordering of the vertices in $G$. Though computing the bandwidth ordering of vertices for a graph is NP-hard, there are heuristics that perform very well in practice. Even theoretically, there is an $O(\log^4 n)$ approximation algorithm for computing the bandwidth ordering of a graph using which our algorithm can produce a $k$-cube representation of any given graph in $k=O(\Delta(\ln b + \ln\ln n))$ dimensions. Both the bounds on cubicity are shown to be tight upto a factor of $O(\log\log n)$., Comment: 12 pages, 0 figures

We give an efficient randomized algorithm to construct a box representation of any graph G on n vertices in $1.5 (\Delta + 2) \ln n$ dimensions, where $\Delta$ is the maximum degree of G. We also show that $\boxi(G) \le (\Delta + 2) \ln n$ for any graph G. Our bound is tight up to a factor of $\ln n$. We also show that our randomized algorithm can be derandomized to get a polynomial time deterministic algorithm. Though our general upper bound is in terms of maximum degree $\Delta$, we show that for almost all graphs on n vertices, its boxicity is upper bound by $c\cdot(d_{av} + 1) \ln n$ where d_{av} is the average degree and c is a small constant. Also, we show that for any graph G, $\boxi(G) \le \sqrt{8 n d_{av} \ln n}$, which is tight up to a factor of $b \sqrt{\ln n}$ for a constant b., Comment: preliminary version appeared in Cocoon 2006