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We prove that every planar graph has a representation using axis-parallel cubes in three dimensions in such a way that there is a cube corresponding to each vertex of the planar graph and two cubes have a non-empty intersection if and only if their corresponding vertices are adjacent. Moreover, when two cubes have a non-empty intersection, they just touch… (More)

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… (More)

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. Kratochvíl and Kuběna [2] asked the question of whether the complements of planar graphs are segment graphs. We show here that the complements of all… (More)

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 χ ′ s (G), is the minimum number of colours needed in any strong edge colouring of G. A graph is said to be… (More)

An axis-parallel d–dimensional box is a Cartesian product R 1 × R 2 × · · · × R d where R i (for 1 ≤ i ≤ d) is a closed interval of the form [a i , b i ] on the real line. For a graph G, its boxicity box(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… (More)

An axis-parallel k-dimensional box is a Cartesian product R 1 × R 2 × · · · × R k where R i (for 1 ≤ i ≤ k) is a closed interval of the form [a i , b i ] on the real line. For a graph G, its boxicity box(G) is the minimum dimension k, such that G is rep-resentable as the intersection graph of (axis-parallel) boxes in k-dimensional space. The concept of… (More)

A unit cube in k dimensional space (or k-cube in short) is defined as the Cartesian product R1 × R2 × · · · × R k where Ri(for 1 ≤ i ≤ k) is a closed interval of the form [ai, ai + 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… (More)

A k-dimensional box is the Cartesian product R1 × R2 × · · · × R k where each Ri 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… (More)

A k-cube (or " a unit cube in k dimensions ") is defined as the Cartesian product R1 ×. .. × R k where Ri(for 1 ≤ i ≤ k) is an interval of the form [ai, ai + 1] on the real line. The k-cube representation of a graph G is a mapping of the vertices of G to k-cubes such that the k-cubes mapped to two vertices in G have a non-empty intersection if and only if… (More)

A circular-arc graph is the intersection graph of arcs of a circle. It is a well-studied graph model with numerous natural applications. A certifying algorithm is an algorithm that outputs a certificate, along with its answer (be it positive or negative), where the certificate can be used to easily justify the given answer. While the recognition of… (More)