172 research outputs found

    BILANGAN KROMATIK HARMONIS PADA GRAF PAYUNG, GRAF PARASUT, DAN GRAF SEMI PARASUT

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    This article discusses the harmonic coloring of simple graphs G, namely umbrella graphs, parachute graphs, and semi-parachute graphs. A vertex coloring on a graph G is a harmonic coloring if each pair of colors (based on edges or pair of vertices) appears at most once. The chromatic number associated with the harmonic coloring of graph G is called the harmonic chromatic number denoted XH(G). In this article, the exact values ​​of harmonic chromatic numbers are obtained for umbrella graphs, parachute graphs, and semi-parachute graphs

    The Edge-Distinguishing Chromatic Number of Petal Graphs, Chorded Cycles, and Spider Graphs

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    The edge-distinguishing chromatic number (EDCN) of a graph GG is the minimum positive integer kk such that there exists a vertex coloring c:V(G){1,2,,k}c:V(G)\to\{1,2,\dotsc,k\} whose induced edge labels {c(u),c(v)}\{c(u),c(v)\} are distinct for all edges uvuv. Previous work has determined the EDCN of paths, cycles, and spider graphs with three legs. In this paper, we determine the EDCN of petal graphs with two petals and a loop, cycles with one chord, and spider graphs with four legs. These are achieved by graph embedding into looped complete graphs.Comment: 23 pages, 1 figur

    Upward Three-Dimensional Grid Drawings of Graphs

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    A \emph{three-dimensional grid drawing} of a graph is a placement of the vertices at distinct points with integer coordinates, such that the straight line segments representing the edges do not cross. Our aim is to produce three-dimensional grid drawings with small bounding box volume. We prove that every nn-vertex graph with bounded degeneracy has a three-dimensional grid drawing with O(n3/2)O(n^{3/2}) volume. This is the broadest class of graphs admiting such drawings. A three-dimensional grid drawing of a directed graph is \emph{upward} if every arc points up in the z-direction. We prove that every directed acyclic graph has an upward three-dimensional grid drawing with (n3)(n^3) volume, which is tight for the complete dag. The previous best upper bound was O(n4)O(n^4). Our main result is that every cc-colourable directed acyclic graph (cc constant) has an upward three-dimensional grid drawing with O(n2)O(n^2) volume. This result matches the bound in the undirected case, and improves the best known bound from O(n3)O(n^3) for many classes of directed acyclic graphs, including planar, series parallel, and outerplanar

    On the Graceful Game

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    A graceful labeling of a graph GG with mm edges consists of labeling the vertices of GG with distinct integers from 00 to mm such that, when each edge is assigned as induced label the absolute difference of the labels of its endpoints, all induced edge labels are distinct. Rosa established two well known conjectures: all trees are graceful (1966) and all triangular cacti are graceful (1988). In order to contribute to both conjectures we study graceful labelings in the context of graph games. The Graceful game was introduced by Tuza in 2017 as a two-players game on a connected graph in which the players Alice and Bob take turns labeling the vertices with distinct integers from 0 to mm. Alice's goal is to gracefully label the graph as Bob's goal is to prevent it from happening. In this work, we study winning strategies for Alice and Bob in complete graphs, paths, cycles, complete bipartite graphs, caterpillars, prisms, wheels, helms, webs, gear graphs, hypercubes and some powers of paths

    Friendly index sets of starlike graphs

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    For a graph G = (V, E) and a coloring (labeling) f : V(G) → Z2 let vf(i) = | f-1(i)|. The coloring f is said to be friendly if |vf(1) - v f(0)| ≤ 1. The coloring f : V( G) → Z2 induces an edge labeling f* : E( G) → Z2 defined by f* (xy) = f( x) + f(y) (mod 2). Let ef(i) = |f*-1( i)|. The friendly index set of the graph G, denoted by FI (G), is defined by FIG= ef1-ef 0:f isafriendly vertexlabelingof G. In this thesis the friendly index sets of certain classes of trees, called starlike graphs, will be determined

    The b-Chromatic Number of Star Graph Families

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    In this paper, we investigate the b-chromatic number of central graph, middle graph and total graph of star graph, denoted by C(K1,n), M(K1,n)  and  T(K1,n) respectively. We discuss the relationship between b-chromatic number with some other types of chromatic numbers such as chromatic number, star chromatic number and equitable chromatic number
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