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.

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.

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.

scholarly journals Spanning $k$-trees of Bipartite Graphs

10.37236/3628 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Mikio Kano ◽  
Kenta Ozeki ◽  
Kazuhiro Suzuki ◽  
Masao Tsugaki ◽  
Tomoki Yamashita
Keyword(s):  
Bipartite Graph ◽  
Connected Graph ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Bipartite Graphs ◽  
Degree Sum

A tree is called a $k$-tree if its maximum degree is at most $k$. We prove the following theorem. Let $k \geq 2$ be an integer, and $G$ be a connected bipartite graph with bipartition $(A,B)$ such that $|A| \le |B| \le (k-1)|A|+1$. If $\sigma_k(G) \ge |B|$, then $G$ has a spanning $k$-tree, where $\sigma_k(G)$ denotes the minimum degree sum of $k$ independent vertices of $G$. Moreover, the condition on $\sigma_k(G)$ is sharp. It was shown by Win (Abh. Math. Sem. Univ. Hamburg, 43, 263–267, 1975) that if a connected graph $H$ satisfies $\sigma_k(H) \ge |H|-1$, then $H$ has a spanning $k$-tree. Thus our theorem shows that the condition becomes much weaker if the graph is bipartite.

On Dirac's Generalization of Brooks' Theorem

1972 ◽  
Vol 24 (5) ◽  
pp. 805-807 ◽  
Author(s):  
Hudson V. Kronk ◽  
John Mitchem
Keyword(s):  
Chromatic Number ◽  
Complete Graph ◽  
Connected Graph ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Equivalent Formulation ◽  
Critical Graphs ◽  
Critical Graph ◽  
Odd Cycle ◽  
Proper Subgraph

It is easy to verify that any connected graph G with maximum degree s has chromatic number χ(G) ≦ 1 + s. In [1], R. L. Brooks proved that χ(G) ≦ s, unless s = 2 and G is an odd cycle or s > 2 and G is the complete graph Ks+1. This was the first significant theorem connecting the structure of a graph with its chromatic number. For s ≦ 4, Brooks' theorem says that every connected s-chromatic graph other than Ks contains a vertex of degree > s — 1. An equivalent formulation can be given in terms of s-critical graphs. A graph G is said to be s-critical if χ(G) = s, but every proper subgraph has chromatic number less than s. Each scritical graph has minimum degree ≦ s — 1. We can now restate Brooks' theorem: if an s-critical graph, s ≦ 4, is not Ks and has p vertices and q edges, then 2q ≦ (s — l)p + 1. Dirac [2] significantly generalized the theorem of Brooks by showing that 2q ≦ (s — 1)£ + s — 3 and that this result is best possible.

scholarly journals Nordhaus–Gaddum-Type Results for the Steiner Gutman Index of Graphs

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1711
Author(s):  
Zhao Wang ◽  
Yaping Mao ◽  
Kinkar Chandra Das ◽  
Yilun Shang
Keyword(s):  
Lower Bounds ◽  
Connected Graph ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Upper And Lower Bounds ◽  
K Index ◽  
Gutman Index

Building upon the notion of the Gutman index SGut(G), Mao and Das recently introduced the Steiner Gutman index by incorporating Steiner distance for a connected graph G. The Steiner Gutman k-index SGutk(G) of G is defined by SGutk(G)=∑S⊆V(G),|S|=k∏v∈SdegG(v)dG(S), in which dG(S) is the Steiner distance of S and degG(v) is the degree of v in G. In this paper, we derive new sharp upper and lower bounds on SGutk, and then investigate the Nordhaus-Gaddum-type results for the parameter SGutk. We obtain sharp upper and lower bounds of SGutk(G)+SGutk(G¯) and SGutk(G)·SGutk(G¯) for a connected graph G of order n, m edges, maximum degree Δ and minimum degree δ.

scholarly journals Bounds for Identifying Codes in Terms of Degree Parameters

10.37236/2036 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Florent Foucaud ◽  
Guillem Perarnau
Keyword(s):  
Large Class ◽  
Connected Graph ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Clique Number ◽  
Minimum Size ◽  
Regular Graphs ◽  
Identifying Codes ◽  
Identifying Code ◽  
Sharp Estimates

An identifying code is a subset of vertices of a graph such that each vertex is uniquely determined by its neighbourhood within the identifying code. If $\gamma^{\text{ID}}(G)$ denotes the minimum size of an identifying code of a graph $G$, it was conjectured by F. Foucaud, R. Klasing, A. Kosowski and A. Raspaud that there exists a constant $c$ such that if a connected graph $G$ with $n$ vertices and maximum degree $d$ admits an identifying code, then $\gamma^{\text{ID}}(G)\leq n-\tfrac{n}{d}+c$. We use probabilistic tools to show that for any $d\geq 3$, $\gamma^{\text{ID}}(G)\leq n-\tfrac{n}{\Theta(d)}$ holds for a large class of graphs containing, among others, all regular graphs and all graphs of bounded clique number. This settles the conjecture (up to constants) for these classes of graphs. In the general case, we prove $\gamma^{\text{ID}}(G)\leq n-\tfrac{n}{\Theta(d^{3})}$. In a second part, we prove that in any graph $G$ of minimum degree $\delta$ and girth at least 5, $\gamma^{\text{ID}}(G)\leq(1+o_\delta(1))\tfrac{3\log\delta}{2\delta}n$. Using the former result, we give sharp estimates for the size of the minimum identifying code of random $d$-regular graphs, which is about $\tfrac{\log d}{d}n$.

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