A monotone path in an edge-ordered graph

Discrete Mathematics, 1994

This paper deals with certain edge labelings of graphs. After having introduced the concepts of a weak antimagic graph and an Egyptian magic graph, the authors showed that every connected graph of order 23 is weakly antimagic and proved some interesting relations between the different edge labelings. * Corresponding author 0012-365X/94/$07.00 0 1994-Elsevier Science B.V. All rights reserved SSDl 0012-365X(93)E0054-8

2012

Let G be a graph with vertex set V(G) and edge set E(G), and let A ={O, I}. A labeling f: V(G)-7 A induces a partial edge labeling f * : E(G)-7 A defined by f*(xy) = f{x), ifand only iff{x) f{y), for each edge xy E E(G). For i E A, let vt<i) = card{v E V(G): f{v) i} and er-(i) card{e E E(G): f"(e) = i}. A labeling f ofa graph G is said to be friendly ifI vt<0) vt< I) I =:;; 1. If, let< 0) et<l) I =:;; I then G is said to be balanced. The balance index set ofthe graph G, Bl(G), is defined as {let<O)- et<1)1: the vertex labeling fis friendly}. Results parallel to the concept offriendly index sets are presented. 1. Introduction. In 1.2], A. Liu, S.K. Tan and the second author considered a new labeling problem of graph theory. Let G be a graph with vertex set V(G) and edge set E(G). A vertex labeling ofG is a mapping ffrom V(G) into the set {O, I}. For each vertex labeling f of G, we can define a partial edge labeling fIiI ofG in the

international symposium on algorithms and computation, 2016

Let G be a (p, q) graph. Let f : V (G) → {1, 2,. .. , k} be a map. For each edge uv, assign the label gcd (f (u), f (v)). f is called k-prime cordial labeling of G if |v f (i) − v f (j)| ≤ 1, i, j ∈ {1, 2,. .. , k} and |e f (0) − e f (1)| ≤ 1 where v f (x) denotes the number of vertices labeled with x, e f (1) and e f (0) respectively denote the number of edges labeled with 1 and not labeled with 1. A graph with a k-prime cordial labeling is called a k-prime cordial graph. In this paper we investigate 4prime cordial labeling behavior of complete graph, book, flower, mC n and some more graphs.

Discrete Mathematics, 1982

European Journal of Combinatorics, 1996

This paper deals with graphs the automorphism groups of which are transitive on vertices and on undirected paths (but not necessarily on directed walks) of some fixed length. In particular , it is shown that if the automorphism group G of a graph ⌫ is transitive on vertices and on undirected paths of length k ϩ 1 in ⌫ , for some k у 1 , then G is also transitive on k-arcs in ⌫. Further details are given for the case k ϭ 1 , for the case of cubic graphs , and for the case k Ͼ 4 .

We prove a decomposition theorem for graphs that do not contain a subdivision of K 4 as an induced subgraph where K 4 is the complete graph on four vertices. We obtain also a structure theorem for the class C of graphs that contain neither a subdivision of K 4 nor a wheel as an induced subgraph, where a wheel is a cycle on at least four vertices together with a vertex that has at least three neighbors on the cycle. Our structure theorem is used to prove that every graph in C is 3-colorable and entails a polynomial-time recognition algorithm for membership in C. As an intermediate result, we prove a structure theorem for the graphs whose cycles are all chordless. q } be the two sides of the bipartition of H . If v is adjacent to at most one vertex in A and at most one in B, then the lemma holds. Suppose now, up to symmetry, that v is adjacent to at least two vertices in A, say a 1 , a 2 . Then v is either adjacent to every vertex in B or to no vertex in B, for otherwise, up to symmetry, v is adjacent to b 1 and not to b 2 , and {a 1 ,

Mathematics and Statistics, 2023

Researchers have constructed a model to transform "word motion problems into an algorithmic form" in order to be processed by an intelligent tutoring system (ITS). This process has the following steps. Step 1: Categorizing the characteristics of motion problems, step 2: suggesting a model for the categories. "In order to solve all categories of problems, graph theory including backward and forward chaining techniques of artificial intelligence can be utilized". The adoption of graph theory into motion problems has evidence that the model solves almost all of motion problems. Graph labeling is sub field of graph theory which has become the area of interest due to its diversified applications. Formally, if the nodes are labeled under some constraint, the resulting labeling is known as vertex labeling and it will be an edge labeling if the labels are assigned to edges under some conditions. Graph labeling nowadays is one of the rapid growing areas in applied mathematics which has shown its presence in almost every field. The known applications are in Computer Science, Physics, Chemistry, Radar, Coding Theory, Connectomics, Socioloy, x-ray crystallography, Astronomy etc. "For a graph G(V, E) and k > 0, give node labels from {0, 1,. .. , k − 1} such that when the edge labels are induced by the absolute value of the difference of the node labels, the count of nodes labeled with i and the count of nodes labeled with j differ by at most one and the number of lines labeled with i and with j differ by at most 1. So G with such an allocation of labels is k−equitable and becomes 3-equitable labeling, when k = 3". In this paper, the existence and non-existence of 3-equitable labeling of certain graphs are established.

2013

Let c be a proper k-coloring of a connected graph G. Let Π = {S 1 , S 2 ,. .. , S k } be the induced partition of V (G) by c, where S i is the partition class having all vertices with color i. The color code c Π (v) of vertex v is the ordered k-tuple (d(v, S 1), d(v, S 2),. .. , d(v, S k)), where d(v, S i) = min{d(v, x)|x ∈ S i }, for 1 ≤ i ≤ k. If all vertices of G have distinct color codes, then c is called a locating-coloring of G. The locating-chromatic number of G, denoted by χ L (G), is the smallest k such that G posses a locating k-coloring. Clearly, any graph of order n ≥ 2 has locating-chromatic number k, where 2 ≤ k ≤ n. Characterizing all graphs with a certain locating-chromatic number is a difficult problem. Up to now, all graphs of order n with locating chromatic number 2, n − 1, or n have been characterized. In this paper, we characterize all trees whose locating-chromatic number is 3. We also give a family of trees with locating-chromatic number 4.

Proyecciones (Antofagasta)

For a connected graph G = (V, E), a monophonic set of G is a set M ⊆ V (G) such that every vertex of G is contained in a monophonic path joining some pair of vertices in M. A subset D of vertices in G is called dominating set if every vertex not in D has at least one neighbour in D. A monophonic dominating set M is both a monophonic and a dominating set. The monophonic,dominating,monophonic domination number m(G), γ(G), γ m (G) respectively are the minimum cardinality of the respective sets in G. Monophonic domination number of certain classes of graphs are determined. Connected graph of order p with monophonic domination number p − 1 or p is characterised. It is shown that for every two intigers a, b ≥ 2 with 2 ≤ a ≤ b, there is a connected graph G such that γ m (G) = a and γ g (G) = b, where γ g (G) is the geodetic domination number of a graph.