scholarly journals Colouring the Petals of a Graph

10.37236/1699 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
David Cariolaro ◽  
Gianfranco Cariolaro
Keyword(s):  
Regular Graph ◽  
Connected Graph ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Petersen Graph ◽  
Empty Graph ◽  
Single Exception ◽  
Class 1

A petal graph is a connected graph $G$ with maximum degree three, minimum degree two, and such that the set of vertices of degree three induces a $2$–regular graph and the set of vertices of degree two induces an empty graph. We prove here that, with the single exception of the graph obtained from the Petersen graph by deleting one vertex, all petal graphs are Class $1$. This settles a particular case of a conjecture of Hilton and Zhao.

The f-Chromatic Index of a Graph Whose f-Core Has Maximum Degree 2

2013 ◽  
Vol 56 (3) ◽  
pp. 449-458 ◽  
Author(s):  
S. Akbari ◽  
M. Chavooshi ◽  
M. Ghanbari ◽  
S. Zare
Keyword(s):  
Connected Graph ◽  
Maximum Degree ◽  
Disjoint Union ◽  
Minimum Degree ◽  
Chromatic Index ◽  
Induced Subgraph ◽  
Class 1 ◽  
Minimum Number

Abstract.Let G be a graph. The minimum number of colors needed to color the edges of G is called the chromatic index of G and is denoted by χ'(G). It is well known that , for any graph G, where Δ(G) denotes the maximum degree of G. A graph G is said to be class 1 if x'(G) = Δ(G) and class 2 if χ'(G) = Δ(G)+1. Also, GΔ is the induced subgraph on all vertices of degree Δ(G). Let f : V(G) → ℕ be a function. An f-coloring of a graph G is a coloring of the edges of E(G) such that each color appears at each vertex v ∊ V(G) at most f (v) times. The minimum number of colors needed to f-color G is called the f-chromatic index of G and is denoted by χ'f (G). It was shown that for every graph , where . A graph G is said to be f -class 1 , and f -class 2, otherwise. Also, GΔf is the induced subgraph of G on . Hilton and Zhao showed that if G has maximum degree two and G is class 2, then G is critical, GΔ is a disjoint union of cycles and δ(G) = Δ(G)–1, where δ(G) denotes the minimum degree of G, respectively. In this paper, we generalize this theorem to f -coloring of graphs. Also, we determine the f -chromatic index of a connected graph G with |GΔf| ≤ 4.

scholarly journals On Decomposing Graphs of Large Minimum Degree into Locally Irregular Subgraphs

10.37236/5173 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Jakub Przybyło
Keyword(s):  
Connected Graph ◽  
Special Class ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Probabilistic Approach ◽  
The Family ◽  
Edge Colourings ◽  
Odd Cycles

A locally irregular graph is a graph whose adjacent vertices have distinct degrees. We say that a graph G can be decomposed into k locally irregular subgraphs if its edge set may be partitioned into k subsets each of which induces a locally irregular subgraph in G. It has been conjectured that apart from the family of exceptions which admit no such decompositions, i.e., odd paths, odd cycles and a special class of graphs of maximum degree 3, every connected graph can be decomposed into 3 locally irregular subgraphs. Using a combination of a probabilistic approach and some known theorems on degree constrained subgraphs of a given graph, we prove this to hold for graphs of minimum degree at least $10^{10}$. This problem is strongly related to edge colourings distinguishing neighbours by the pallets of their incident colours and to the 1-2-3 Conjecture. In particular, the contribution of this paper constitutes a strengthening of a result of Addario-Berry, Aldred, Dalal and Reed [J. Combin. Theory Ser. B 94 (2005) 237-244].

scholarly journals Bounds for symmetric division deg index of graphs

Filomat ◽  
2019 ◽  
Vol 33 (3) ◽  
pp. 683-698 ◽  
Author(s):  
Kinkar Das ◽  
Marjan Matejic ◽  
Emina Milovanovic ◽  
Igor Milovanovic
Keyword(s):  
Connected Graph ◽  
Maximum Degree ◽  
Topological Index ◽  
Minimum Degree ◽  
Topological Indices ◽  
Symmetric Division ◽  
Mathematical Properties ◽  
Vertex Degrees ◽  
Additive Modeling ◽  
Simple Connected Graph

LetG = (V,E) be a simple connected graph of order n (?2) and size m, where V(G) = {1, 2,..., n}. Also let ? = d1 ? d2 ?... ? dn = ? > 0, di = d(i), be a sequence of its vertex degrees with maximum degree ? and minimum degree ?. The symmetric division deg index, SDD, was defined in [D. Vukicevic, Bond additive modeling 2. Mathematical properties of max-min rodeg index, Croat. Chem. Acta 83 (2010) 261- 273] as SDD = SDD(G) = ?i~j d2i+d2j/didj, where i~j means that vertices i and j are adjacent. In this paper we give some new bounds for this topological index. Moreover, we present a relation between topological indices of graph.

Longest Cycles in 2-Connected Graphs with Prescribed Maximum Degree

1980 ◽  
Vol 32 (6) ◽  
pp. 1325-1332 ◽  
Author(s):  
J. A. Bondy ◽  
R. C. Entringer
Keyword(s):  
Connected Graph ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Simple Graph ◽  
General Context ◽  
Connected Graphs ◽  
Longest Cycles ◽  
Image Position ◽  
The Relationship

The relationship between the lengths of cycles in a graph and the degrees of its vertices was first studied in a general context by G. A. Dirac. In [5], he proved that every 2-connected simple graph on n vertices with minimum degree d contains a cycle of length at least min{2d, n};. Dirac's theorem was subsequently strengthened in various directions in [7], [6], [13], [12], [2], [1], [11], [8], [14], [15] and [16].Our aim here is to investigate another aspect of this relationship, namely how the lengths of the cycles in a 2-connected graph depend on the maximum degree. Let us denote by ƒ(n, d) the largest integer k such that every 2-connected simple graph on n vertices with maximum degree d contains a cycle of length at least k. We prove in Section 2 that, for d ≧ 3 and n ≧ d + 2,

scholarly journals Wiener index in graphs with given minimum degree and maximum degree

2021 ◽  
Vol vol. 23 no. 1 (Graph Theory) ◽  
Author(s):  
Peter Dankelmann ◽  
Alex Alochukwu
Keyword(s):  
Graph Theory ◽  
Upper Bound ◽  
Connected Graph ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Large Degree ◽  
Wiener Index ◽  
Sharp Bound

Let $G$ be a connected graph of order $n$.The Wiener index $W(G)$ of $G$ is the sum of the distances between all unordered pairs of vertices of $G$. In this paper we show that the well-known upper bound $\big( \frac{n}{\delta+1}+2\big) {n \choose 2}$ on the Wiener index of a graph of order $n$ and minimum degree $\delta$ [M. Kouider, P. Winkler, Mean distance and minimum degree. J. Graph Theory 25 no. 1 (1997)] can be improved significantly if the graph contains also a vertex of large degree. Specifically, we give the asymptotically sharp bound $W(G) \leq {n-\Delta+\delta \choose 2} \frac{n+2\Delta}{\delta+1}+ 2n(n-1)$ on the Wiener index of a graph $G$ of order $n$, minimum degree $\delta$ and maximum degree $\Delta$. We prove a similar result for triangle-free graphs, and we determine a bound on the Wiener index of $C_4$-free graphs of given order, minimum and maximum degree and show that it is, in some sense, best possible.

scholarly journals Identifying Vertex Covers in Graphs

10.37236/2114 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Michael A Henning ◽  
Anders Yeo
Keyword(s):  
Regular Graph ◽  
Upper Bound ◽  
Maximum Degree ◽  
Vertex Cover ◽  
Nonempty Intersection ◽  
Petersen Graph ◽  
Minimum Size ◽  
Regular Graphs ◽  
Packing Number

An identifying vertex cover in a graph $G$ is a subset $T$ of vertices in $G$ that has a nonempty intersection with every edge of $G$ such that $T$ distinguishes the edges, that is, $e \cap T \ne \emptyset$ for every edge $e$ in $G$ and $e \cap T \ne f \cap T$ for every two distinct edges $e$ and $f$ in $G$. The identifying vertex cover number $\tau_D(G)$ of $G$ is the minimum size of an identifying vertex cover in $G$. We observe that $\tau_D(G) + \rho(G) = |V(G)|$, where $\rho(G)$ denotes the packing number of $G$. We conjecture that if $G$ is a graph of order $n$ and size $m$ with maximum degree $\Delta$, then $\tau_D(G) \le \left( \frac{\Delta(\Delta - 1)}{\Delta^2 + 1} \right) n + \left( \frac{2}{\Delta^2 + 1} \right) m$. If the conjecture is true, then the bound is best possible for all $\Delta \ge 1$. We prove this conjecture when $\Delta \ge 1$ and $G$ is a $\Delta$-regular graph. The three known Moore graphs of diameter two, namely the $5$-cycle, the Petersen graph and the Hoffman-Singleton graph, are examples of regular graphs that achieves equality in the upper bound. We also prove this conjecture when $\Delta \in \{2,3\}$.

Excluding Minors in Nonplanar Graphs of Girth at Least Five

2000 ◽  
Vol 9 (6) ◽  
pp. 573-585 ◽  
Author(s):  
ROBIN THOMAS ◽  
JAN McDONALD THOMSON
Keyword(s):  
Connected Graph ◽  
Minimum Degree ◽  
The Other ◽  
Petersen Graph ◽  
Regular Graphs ◽  
Nonplanar Graph ◽  
A Minor

A graph G is quasi 4-connected if it is simple, 3-connected, has at least five vertices, and for every partition (A, B, C) of V(G) either [mid ]C[mid ] [ges ] 4, or G has an edge with one end in A and the other end in B, or one of A,B has at most one vertex. We show that any quasi 4-connected nonplanar graph with minimum degree at least three and no cycle of length less than five has a minor isomorphic to P−10, the Petersen graph with one edge deleted. We deduce the following weakening of Tutte's Four Flow Conjecture: every 2-edge-connected graph with no minor isomorphic to P−10 has a nowhere-zero 4-flow. This extends a result of Kilakos and Shepherd who proved the same for 3-regular graphs.

Radius, Diameter and the Degree Sequence of a Graph

2015 ◽  
Vol 65 (6) ◽  
Author(s):  
Jaya Percival Mazorodze ◽  
Simon Mukwembi
Keyword(s):  
Connected Graph ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Degree Sequence ◽  
Upper Bounds ◽  
Free Graph

AbstractWe give asymptotically sharp upper bounds on the radius and diameter of(i) a connected graph,(ii) a connected triangle-free graph,(iii) a connected C4-free graph of given order, minimum degree, and maximum degree.We also give better bounds on the radius and diameter for triangle-free graphs with a given order, minimum degree and a given number of distinct terms in the degree sequence of the graph. Our results improve on old classical theorems by Erd˝os, Pach, Pollack and Tuza [Radius, diameter, and minimum degree, J. Combin. Theory Ser. B 47 (1989), 73-79] on radius, diameter and minimum degree.

scholarly journals A degree condition for the existence ofk-factors with prescribed properties

2005 ◽  
Vol 2005 (6) ◽  
pp. 863-873 ◽  
Author(s):  
Changping Wang
Keyword(s):  
Connected Graph ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Degree Condition

Letkbe an integer such thatk≥3, and letGbe a 2-connected graph of ordernwithn≥4k+1,kneven, and minimum degree at leastk+1. We prove that if the maximum degree of each pair of nonadjacent vertices is at leastn/2, thenGhas ak-factor excluding any given edge. The result of Nishimura (1992) is improved.

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