Solutions of some L(2,1)-coloring related open problems

Mandal, Nibedita; Panigrahi, Pratima

doi:10.7151/dmgt.1858

Cited by 4 publications

(4 citation statements)

References 2 publications

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“…If the tree 2 with at least five major vertices is a subtree of a tree with maximum degree same as that of 2 , then ( ) = 0. 4 , and V 5 be five major vertices adjacent to and receive the colors 1 , 2 , 3 , 4 , and 5 , respectively, by a one-hole span coloring of 2 . Without loss of generality, we assume that 0 ⩽ 1 < 2 < 3 < 4 < 5 ⩽ Δ + 2.…”

Section: Infinitely Many Trees With Holes 0 1 Andmentioning

confidence: 99%

“…Zhai et al [3] have improved the above condition as a tree with no pair of major vertices at distances 2 and 4 is Type-I. Mandal and Panigrahi [4] have proved that ( ) = Δ + 1 if has at most one pair of major vertices at distance either 2 or 4 and all other pairs are at distance at least 7. Wood and Jacob [5] have given a complete characterization of the (2, 1)-span of trees up to twenty vertices.…”

Section: Introductionmentioning

confidence: 99%

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Infinitely Many Trees with Maximum Number of Holes Zero, One, and Two

Kola

Gudla

Niranjan

2018

Journal of Applied Mathematics

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An L(2,1)-coloring of a simple connected graph G is an assignment f of nonnegative integers to the vertices of G such that fu-fv⩾2 if d(u,v)=1 and fu-fv⩾1 if d(u,v)=2 for all u,v∈V(G), where d(u,v) denotes the distance between u and v in G. The span of f is the maximum color assigned by f. The span of a graph G, denoted by λ(G), is the minimum of span over all L(2,1)-colorings on G. An L(2,1)-coloring of G with span λ(G) is called a span coloring of G. An L(2,1)-coloring f is said to be irreducible if there exists no L(2,1)-coloring g such that g(u)⩽f(u) for all u∈V(G) and g(v)<f(v) for some v∈V(G). If f is an L(2,1)-coloring with span k, then h∈0,1,2,…,k is a hole if there is no v∈V(G) such that f(v)=h. The maximum number of holes over all irreducible span colorings of G is denoted by Hλ(G). A tree T with maximum degree Δ having span Δ+1 is referred to as Type-I tree; otherwise it is Type-II. In this paper, we give a method to construct infinitely many trees with at least one hole from a one-hole tree and infinitely many two-hole trees from a two-hole tree. Also, using the method, we construct infinitely many Type-II trees with maximum number of holes one and two. Further, we give a sufficient condition for a Type-II tree with maximum number of holes zero.

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Section: Infinitely Many Trees With Holes 0 1 Andmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Infinitely Many Trees with Maximum Number of Holes Zero, One, and Two

Kola

Gudla

Niranjan

2018

Journal of Applied Mathematics

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“…Zhai et al [3] improved the above condition as if T does not contains ∆-path segment of length 2 and 4, then λ(T ) = ∆ + 1. Mandal and Panigrahi [4] have found that λ(T ) = ∆ + 1 if T has at most one ∆-path segment of length either 2 or 4 and all other ∆-path segments are of length at least 7. Wood and Jacob [5] have given a complete characterization of the L(2, 1)-span of trees up to twenty vertices.…”

Section: Introductionmentioning

confidence: 99%

Some classes of trees with maximum number of holes two

Kola

Gudla

Niranjan

2020

AKCE International Journal of Graphs and Combinatorics

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An L (2, 1)-coloring of a simple connected graph G is an assignment of non-negative integers to the vertices of G such that adjacent vertices color difference is at least two, and vertices that are at distance two from each other get different colors. The maximum color assigned in an L (2, 1)-coloring is called span of that coloring. The span of a graph G denoted by λ (G) is the smallest span taken over all L (2, 1)-colorings of G. A hole is an unused color within the range of colors used by the coloring. An L (2, 1)-coloring f is said to be irreducible if no other L (2, 1)-coloring can be produced by decreasing a color of f . The maximum number of holes of a graph G, denoted by H λ (G), is the maximum number of holes taken over all irreducible L (2, 1)-colorings with span λ (G). Laskar and Eyabi (Christpher, 2009) conjectured that if T is a tree, then H λ (T ) = 2 if and only if T = P n , n > 4. We show that this conjecture does not hold by providing a counterexample. Also, we give some classes of trees with maximum number of holes two.

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“…4, m ! 2, k inh ðK n w P m Þ ¼ 2n À 1: Mandal and Panigrahi [12] solved some open problems related to irreducible no-hole coloring of graphs. The same authors also studied inh-colorability of hypercubes [13], subdivision of graphs [14], and Cartesian product of a complete graph and a cycle [15].…”

Section: Introductionmentioning

confidence: 99%

On irreducible no-hole L(2, 1)-coloring of Cartesian product of trees with paths

Mandal

Panigrahi

2020

AKCE International Journal of Graphs and Combinatorics

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An L(2, 1)-coloring of a graph G is a mapping f : VðGÞ ! Z þ [ f0g such that jf ðuÞ À f ðvÞj ! 2 for all edges uv of G, and jf ðuÞ À f ðvÞj ! 1 if u and v are at distance two in G. The span of an L(2, 1)coloring f of G, denoted by span(f), is max ff ðvÞ : v 2 VðGÞg: The span of G, denoted by kðGÞ, is the minimum span of all possible L(2, 1)-colorings of G. If f is an L(2, 1)-coloring of a graph G with span k then an integer l is a hole in f if l 2 ð0, kÞ and there is no vertex v in G such that f(v) ¼ l. A no-hole coloring is defined to be an L(2, 1)-coloring with no hole in it. An L(2, 1)-coloring is said to be irreducible if the color of none of the vertices in the graph can be decreased and yield another L(2, 1)-coloring of the same graph. An irreducible no-hole coloring of a graph G, in short inh-coloring of G, is an L(2, 1)-coloring of G which is both irreducible and no-hole. A graph G is inh-colorable if there exists an inh-coloring of it. For an inh-colorable graph G the lower inh-span or simply inh-span of G, denoted by k inh ðGÞ, is defined as k inh ðGÞ ¼ minfspan ðf Þ : f is an inh-coloring of G}. In this paper, we prove that the Cartesian product of trees with paths are inh-colorable. KEYWORD L(2, 1)-coloring; no-hole coloring; irreducible coloring; span of a graph; Cartesian product of graphs 2010 MSC

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Solutions of some L(2,1)-coloring related open problems

Cited by 4 publications

References 2 publications

Infinitely Many Trees with Maximum Number of Holes Zero, One, and Two

Infinitely Many Trees with Maximum Number of Holes Zero, One, and Two

Some classes of trees with maximum number of holes two

On irreducible no-hole L(2, 1)-coloring of Cartesian product of trees with paths

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