scholarly journals Distant Set Distinguishing Total Colourings of Graphs

10.37236/5481 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Jakub Przybyło
Keyword(s):  
Upper Bound ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Probabilistic Method ◽  
Constant Factor ◽  
Additive Constant ◽  
Regular Graphs

The Total Colouring Conjecture suggests that $\Delta+3$ colours ought to suffice in order to provide a proper total colouring of every graph $G$ with maximum degree $\Delta$. Thus far this has been confirmed up to an additive constant factor, and the same holds even if one additionally requires every pair of neighbours in $G$ to differ with respect to the sets of their incident colours, so called pallets. Within this paper we conjecture that an upper bound of the form $\Delta+C$, for a constant $C>0$ still remains valid even after extending the distinction requirement to pallets associated with vertices at distance at most $r$, if only $G$ has minimum degree $\delta$ larger than a constant dependent on $r$. We prove that such assumption on $\delta$ is then unavoidable and exploit the probabilistic method in order to provide two supporting results for the conjecture. Namely, we prove the upper bound $(1+o(1))\Delta$ for every $r$, and show that for any fixed $\epsilon\in(0,1]$ and $r$, the conjecture holds if $\delta\geq \varepsilon\Delta$, i.e., in particular for regular graphs.

scholarly journals The Diameter of Almost Eulerian Digraphs

10.37236/429 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Peter Dankelmann ◽  
L. Volkmann
Keyword(s):  
Graph Theory ◽  
Directed Graph ◽  
Upper Bound ◽  
Minimum Degree ◽  
Additive Constant ◽  
Undirected Graphs

Soares [J. Graph Theory 1992] showed that the well known upper bound $\frac{3}{\delta+1}n+O(1)$ on the diameter of undirected graphs of order $n$ and minimum degree $\delta$ also holds for digraphs, provided they are eulerian. In this paper we investigate if similar bounds can be given for digraphs that are, in some sense, close to being eulerian. In particular we show that a directed graph of order $n$ and minimum degree $\delta$ whose arc set can be partitioned into $s$ trails, where $s\leq \delta-2$, has diameter at most $3 ( \delta+1 - \frac{s}{3})^{-1}n+O(1)$. If $s$ also divides $\delta-2$, then we show the diameter to be at most $3(\delta+1 - \frac{(\delta-2)s}{3(\delta-2)+s} )^{-1}n+O(1)$. The latter bound is sharp, apart from an additive constant. As a corollary we obtain the sharp upper bound $3( \delta+1 - \frac{\delta-2}{3\delta-5})^{-1} n + O(1)$ on the diameter of digraphs that have an eulerian trail.

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 A New Upper Bound on the Total Domination Number of a Graph

10.37236/983 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Michael A. Henning ◽  
Anders Yeo
Keyword(s):  
Graph Theory ◽  
Upper Bound ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Total Domination ◽  
Total Domination Number ◽  
Minimum Cardinality ◽  
Total Dominating Set

A set $S$ of vertices in a graph $G$ is a total dominating set of $G$ if every vertex of $G$ is adjacent to some vertex in $S$. The minimum cardinality of a total dominating set of $G$ is the total domination number of $G$. Let $G$ be a connected graph of order $n$ with minimum degree at least two and with maximum degree at least three. We define a vertex as large if it has degree more than $2$ and we let ${\cal L}$ be the set of all large vertices of $G$. Let $P$ be any component of $G - {\cal L}$; it is a path. If $|P| \equiv 0 \, ( {\rm mod} \, 4)$ and either the two ends of $P$ are adjacent in $G$ to the same large vertex or the two ends of $P$ are adjacent to different, but adjacent, large vertices in $G$, we call $P$ a $0$-path. If $|P| \ge 5$ and $|P| \equiv 1 \, ( {\rm mod} \, 4)$ with the two ends of $P$ adjacent in $G$ to the same large vertex, we call $P$ a $1$-path. If $|P| \equiv 3 \, ( {\rm mod} \, 4)$, we call $P$ a $3$-path. For $i \in \{0,1,3\}$, we denote the number of $i$-paths in $G$ by $p_i$. We show that the total domination number of $G$ is at most $(n + p_0 + p_1 + p_3)/2$. This result generalizes a result shown in several manuscripts (see, for example, J. Graph Theory 46 (2004), 207–210) which states that if $G$ is a graph of order $n$ with minimum degree at least three, then the total domination of $G$ is at most $n/2$. It also generalizes a result by Lam and Wei stating that if $G$ is a graph of order $n$ with minimum degree at least two and with no degree-$2$ vertex adjacent to two other degree-$2$ vertices, then the total domination of $G$ is at most $n/2$.

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 $\Delta \in \{2,3\}$.

scholarly journals The Random Graph Intuition for the Tournament Game

2015 ◽  
Vol 25 (1) ◽  
pp. 76-88 ◽  
Author(s):  
DENNIS CLEMENS ◽  
HEIDI GEBAUER ◽  
ANITA LIEBENAU
Keyword(s):  
Lower Bound ◽  
Random Graph ◽  
Complete Graph ◽  
Upper Bound ◽  
Winning Strategy ◽  
Constant Factor ◽  
Additive Constant ◽  
Underlying Graph ◽  
Straightforward Application

In the tournament game two players, called Maker and Breaker, alternately take turns in claiming an unclaimed edge of the complete graph Kn and selecting one of the two possible orientations. Before the game starts, Breaker fixes an arbitrary tournament Tk on k vertices. Maker wins if, at the end of the game, her digraph contains a copy of Tk; otherwise Breaker wins. In our main result, we show that Maker has a winning strategy for k = (2 − o(1))log2n, improving the constant factor in previous results of Beck and the second author. This is asymptotically tight since it is known that for k = (2 − o(1))log2n Breaker can prevent the underlying graph of Maker's digraph from containing a k-clique. Moreover, the precise value of our lower bound differs from the upper bound only by an additive constant of 12.We also discuss the question of whether the random graph intuition, which suggests that the threshold for k is asymptotically the same for the game played by two ‘clever’ players and the game played by two ‘random’ players, is supported by the tournament game. It will turn out that, while a straightforward application of this intuition fails, a more subtle version of it is still valid.Finally, we consider the orientation game version of the tournament game, where Maker wins the game if the final digraph – also containing the edges directed by Breaker – possesses a copy of Tk. We prove that in that game Breaker has a winning strategy for k = (4 + o(1))log2n.

scholarly journals Perfect Matchings in $\epsilon$-regular Graphs

10.37236/1351 ◽  
1998 ◽  
Vol 5 (1) ◽  
Author(s):  
Noga Alon ◽  
Vojtech Rödl ◽  
Andrzej Ruciński
Keyword(s):  
Bipartite Graph ◽  
Regular Graph ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Perfect Matchings ◽  
Regular Graphs

A super $(d,\epsilon)$-regular graph on $2n$ vertices is a bipartite graph on the classes of vertices $V_1$ and $V_2$, where $|V_1|=|V_2|=n$, in which the minimum degree and the maximum degree are between $ (d-\epsilon)n$ and $ (d+\epsilon) n$, and for every $U \subset V_1, W \subset V_2$ with $|U| \geq \epsilon n$, $|W| \geq \epsilon n$, $|{{e(U,W) }\over{|U||W|}}-{{e(V_1,V_2)}\over{|V_1||V_2|}}| < \epsilon.$ We prove that for every $1>d >2 \epsilon >0$ and $n>n_0(\epsilon)$, the number of perfect matchings in any such graph is at least $(d-2\epsilon)^n n!$ and at most $(d+2 \epsilon)^n n!$. The proof relies on the validity of two well known conjectures for permanents; the Minc conjecture, proved by Brégman, and the van der Waerden conjecture, proved by Falikman and Egorichev.

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 $\tfrac{\log d}{d}n$.

scholarly journals Every 3-polytope with minimum degree 5 has a 6-cycle with maximum degree at most 11

2014 ◽  
Vol 315-316 ◽  
pp. 128-134 ◽  
Author(s):  
O.V. Borodin ◽  
A.O. Ivanova ◽  
A.V. Kostochka
Keyword(s):  
Maximum Degree ◽  
Minimum Degree

Semi-Deterministic Construction of Scale-Free Networks with Designated Parameters

2018 ◽  
Vol 18 (01) ◽  
pp. 1850001
Author(s):  
NAOKI TAKEUCHI ◽  
SATOSHI FUJITA
Keyword(s):  
Degree Distribution ◽  
Interconnection Networks ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Scale Free ◽  
Scale Free Networks ◽  
Ideal Number ◽  
Message Propagation ◽  
Deterministic Construction ◽  
The Ideal

Scale-free networks have several favorable properties as the topology of interconnection networks such as the short diameter and the quick message propagation. In this paper, we propose a method to construct scale-free networks in a semi-deterministic manner. The proposed algorithm extends the Bulut's algorithm for constructing scale-free networks with designated minimum degree k and maximum degree m, in such a way that: (1) it determines the ideal number of edges derived from the ideal degree distribution; and (2) after connecting each new node to k existing nodes as in the Bulut’s algorithm, it adjusts the number of edges to the ideal value by conducting add/removal of edges. We prove that such an adjustment is always possible if the number of nodes in the network exceeds [Formula: see text]. The performance of the algorithm is experimentally evaluated.

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].

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