Soumen Nandi

On L(k,k−1,…,1) labeling of triangular lattice

Abstract An n - L ( k , k − 1 , … , 1 ) labeling of a simple graph G is a mapping f : V ( G ) → { 0 , 1 , … , n } such that | f ( u ) − f ( v ) | ≥ k + 1 − d ( u , v ) , for all u , v ∈ V ( G ) , where d ( u , v ) is the length of the shortest path connecting u and v . The L ( k , k − 1 , … , 1 ) labeling span λ k ( F ) of a family of graphs F is the minimum n for which each G ∈ F admits an n - L ( k , k − 1 , … , 1 ) labeling. For the family L 3 of all subgraphs of an infinite triangular lattice we provide upper and lower bounds of λ k ( L 3 ) for general k and show that the ratio of the upper and lower bound is at most 4 3 . The upper bound is given by providing an assignment algorithm to the vertices of the infinite triangular lattice.

On Oriented L(p, 1)-labeling

An oriented graph is a directed graph without any directed cycle of length at most 2. In this article, we characterize the oriented L(p, 1)-labeling span \(\lambda ^{o}_{p, 1}(\overrightarrow{G})\) of an oriented graph \(\overrightarrow{G}\) using graph homomorphisms. Using this characterization and probabilistic techniques we prove the upper bound of \(\lambda ^{o}_{p, 1}(\mathcal {G}_{\varDelta }) \le 2.\varDelta ^{2}.2^{\varDelta } + (p\varDelta )\), where \(\mathcal {G}_{\varDelta }\) is the family of graphs with maximum degree at most \(\varDelta \). Moreover, by proving a lower bound exponential in \(\varDelta \) for the same graph family we conclude that the upper bound is not too far from being optimal. We also settle an open problem given by Sen (DMGT 2014) for the family of outerplanar graphs \(\mathcal {O}\) by showing \(\lambda ^{o}_{2, 1}(\mathcal {O}) = 10\).

Computing the Triangle Maximizing the Length of Its Smallest Side Inside a Convex Polygon

Given a convex polygon with n vertices, we study the problem of identifying a triangle with its smallest side as large as possible among all the triangles that can be drawn inside the polygon. We show that at least one of the vertices of such a triangle must be common with a vertex of the polygon. Next we propose an \(O(n^2\log n)\) time algorithm to compute such a triangle inside the given convex polygon.

On Chromatic Number of Colored Mixed Graphs

An (m, n)-colored mixed graph G is a graph with its arcs having one of the m different colors and edges having one of the n different colors. A homomorphism f of an (m, n)-colored mixed graph G to an (m, n)-colored mixed graph H is a vertex mapping such that if uv is an arc (edge) of color c in G, then f (u)f (v) is an arc (edge) of color c in H. The (m,n)-colored mixed chromatic number x((m,n))(G) of an (m, n)-colored mixed graph G is the order (number of vertices) of a smallest homomorphic image of G. This notion was introduced by Nektfil and Raspaud (2000, J. Combin. Theory, Ser. B 80, 147-155). They showed that x((m,n))(G) <= k(2m n)(k-1) where G is a acyclic k-colorable graph. We prove the tightness of this bound. We also show that the acyclic chromatic number of a graph is bounded by k(2) k(2+) inverted left perpedicular log((2m+n)) log((2m+n)) k inverted right perpendicular if its (m, n)-colored mixed chromatic number is at most k. Furthermore, using probabilistic method, we show that for connected graphs with maximum degree its (m, n)-colored mixed chromatic number is at most 2(Delta-1)(2m+n) (2m + n)(Delta-min(2m+m3)+2) In particular, the last result directly improves the upper bound of 2 Delta(2)2(Delta) oriented chromatic number of graphs with maximum degree Delta, obtained by Kostochka et al. J. Graph Theory 24, 331-340) to 2(Delta-1)(2)2(Delta). We also show that there exists a connected graph with maximum degree Delta and (m, n)-colored mixed chromatic number at least (2m + n)(Delta/2).

A lower bound technique for radio k-coloring

An n-radiok-coloring of a graph G is a function l:V(G){0,1,,n} satisfying the condition |l(u)l(v)|k+1d(u,v) for all distinct u,vV(G). The radiok-chromatic numberrck(G) of G is the minimum n such that G admits an n-radio k-coloring. We establish a general technique for computing the lower bound for rck(G) of a general graph G and derive a formula for it. Using this formula we compute lower bounds of rck() for several graphs and provide alternative short proofs for a number of previously established tight lower bounds. In particular, we progress on a conjecture by Kchikech, Khennoufa and Togni (DMGT 2007) regarding rck() of infinite paths and reprove a result by Liu and Zhu (SIDMA 2005). We also provide a 43-approximation algorithm for radio labeling regular grid graph of degree 6.

On oriented cliques with respect to push operation

To push a vertex

Optimal L(3,2,1)-labeling of triangular lattice

v

Chromatic number of signed graphs with bounded maximum degree

of a directed graph

On relative clique number of colored mixed graphs

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Erratum for "On oriented cliques with respect to push operation"

is to change the orientations of all the arcs incident with