scholarly journals Edge-Distinguishing of Star-Free Graphs

10.37236/8882 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Aleksandra Gorzkowska ◽  
Ernest Kargul ◽  
Szymon Musiał ◽  
Katarzyna Pal
Keyword(s):  
Free Graph ◽  
General Graphs ◽  
Edge Colouring

The distinguishing index $D'(G)$ of a graph $G$ is the least number $k$ such that $G$ has an edge colouring with $k$ colours that is only preserved by the trivial automorphism. Pilśniak proved that a connected, claw-free graph has the distingushing index at most three. In this paper, we show that the distingushing index of a connected, claw-free graph with at least six vertices is bounded from above by two. We also consider more general graphs in this sense. Namely, we prove that if $G$ is a connected, $K_{1,s}$-free graph of order at least six, then $D'(G) \leq s-1$.

scholarly journals Edge and Total Choosability of Near-Outerplanar Graphs

10.37236/1124 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Timothy J. Hetherington ◽  
Douglas R. Woodall
Keyword(s):  
Maximum Degree ◽  
Outerplanar Graphs ◽  
Free Graph ◽  
Minor Free Graphs ◽  
Straightforward Consequence ◽  
Edge Colouring

It is proved that, if $G$ is a $K_4$-minor-free graph with maximum degree $\Delta \ge 4$, then $G$ is totally $(\Delta+1)$-choosable; that is, if every element (vertex or edge) of $G$ is assigned a list of $\Delta+1$ colours, then every element can be coloured with a colour from its own list in such a way that every two adjacent or incident elements are coloured with different colours. Together with other known results, this shows that the List-Total-Colouring Conjecture, that ${\rm ch}"(G) = \chi"(G)$ for every graph $G$, is true for all $K_4$-minor-free graphs. The List-Edge-Colouring Conjecture is also known to be true for these graphs. As a fairly straightforward consequence, it is proved that both conjectures hold also for all $K_{2,3}$-minor free graphs and all $(\bar K_2 + (K_1 \cup K_2))$-minor-free graphs.

scholarly journals Anagram-Free Graph Colouring

10.37236/6267 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Tim E. Wilson ◽  
David R. Wood
Keyword(s):  
Chromatic Number ◽  
Lower Bounds ◽  
Maximum Degree ◽  
Empty Word ◽  
Upper And Lower Bounds ◽  
Graph Colouring ◽  
Free Graph ◽  
Random Structures ◽  
Edge Colouring

An anagram is a word of the form $WP$ where $W$ is a non-empty word and $P$ is a permutation of $W$. We study anagram-free graph colouring and give bounds on the chromatic number. Alon et al.[Random Structures & Algorithms 2002] asked whether anagram-free chromatic number is bounded by a function of the maximum degree. We answer  this question in the negative by constructing graphs with maximum degree 3 and unbounded anagram-free chromatic number. We also prove upper and lower bounds on the anagram-free chromatic number of trees in terms of their radius and pathwidth. Finally, we explore extensions to edge colouring and $k$-anagram-free colouring.

scholarly journals Toughness, Forbidden Subgraphs and Pancyclicity

2021 ◽  
Vol 37 (3) ◽  
pp. 839-866
Author(s):  
Wei Zheng ◽  
Hajo Broersma ◽  
Ligong Wang
Keyword(s):  
Recent Article ◽  
Forbidden Subgraph ◽  
Forbidden Subgraphs ◽  
Free Graph ◽  
Induced Subgraph ◽  
Complete Answer ◽  
Open Case

AbstractMotivated by several conjectures due to Nikoghosyan, in a recent article due to Li et al., the aim was to characterize all possible graphs H such that every 1-tough H-free graph is hamiltonian. The almost complete answer was given there by the conclusion that every proper induced subgraph H of $$K_1\cup P_4$$ K 1 ∪ P 4 can act as a forbidden subgraph to ensure that every 1-tough H-free graph is hamiltonian, and that there is no other forbidden subgraph with this property, except possibly for the graph $$K_1\cup P_4$$ K 1 ∪ P 4 itself. The hamiltonicity of 1-tough $$K_1\cup P_4$$ K 1 ∪ P 4 -free graphs, as conjectured by Nikoghosyan, was left there as an open case. In this paper, we consider the stronger property of pancyclicity under the same condition. We find that the results are completely analogous to the hamiltonian case: every graph H such that any 1-tough H-free graph is hamiltonian also ensures that every 1-tough H-free graph is pancyclic, except for a few specific classes of graphs. Moreover, there is no other forbidden subgraph having this property. With respect to the open case for hamiltonicity of 1-tough $$K_1\cup P_4$$ K 1 ∪ P 4 -free graphs we give infinite families of graphs that are not pancyclic.

scholarly journals Scaling Algorithms for Weighted Matching in General Graphs

2018 ◽  
Vol 14 (1) ◽  
pp. 1-35 ◽  
Author(s):  
Ran Duan ◽  
Seth Pettie ◽  
Hsin-Hao Su
Keyword(s):  
Weighted Matching ◽  
Scaling Algorithms ◽  
General Graphs

scholarly journals Universal and unavoidable graphs

2021 ◽  
pp. 1-14
Author(s):  
Matija Bucić ◽  
Nemanja Draganić ◽  
Benny Sudakov
Keyword(s):  
Free Graph ◽  
Turán Number ◽  
Work Done

Abstract The Turán number ex(n, H) of a graph H is the maximal number of edges in an H-free graph on n vertices. In 1983, Chung and Erdős asked which graphs H with e edges minimise ex(n, H). They resolved this question asymptotically for most of the range of e and asked to complete the picture. In this paper, we answer their question by resolving all remaining cases. Our result translates directly to the setting of universality, a well-studied notion of finding graphs which contain every graph belonging to a certain family. In this setting, we extend previous work done by Babai, Chung, Erdős, Graham and Spencer, and by Alon and Asodi.

Distributed garbage collection for general graphs

2018 ◽  
Vol 53 (5) ◽  
pp. 29-44
Author(s):  
Steven R. Brandt ◽  
Hari Krishnan ◽  
Costas Busch ◽  
Gokarna Sharma
Keyword(s):  
Garbage Collection ◽  
Distributed Garbage Collection ◽  
General Graphs

scholarly journals Secure the IoT Networks as Epidemic Containment Game

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 156
Author(s):  
Juntao Zhu ◽  
Hong Ding ◽  
Yuchen Tao ◽  
Zhen Wang ◽  
Lanping Yu
Keyword(s):  
Heuristic Algorithms ◽  
Heuristic Method ◽  
Exact Algorithms ◽  
Epidemic Threshold ◽  
Solution Quality ◽  
Sis Model ◽  
Linear Programming Formulation ◽  
Iot Devices ◽  
General Graphs ◽  
The Internet Of Things

The spread of a computer virus among the Internet of Things (IoT) devices can be modeled as an Epidemic Containment (EC) game, where each owner decides the strategy, e.g., installing anti-virus software, to maximize his utility against the susceptible-infected-susceptible (SIS) model of the epidemics on graphs. The EC game’s canonical solution concepts are the Minimum/Maximum Nash Equilibria (MinNE/MaxNE). However, computing the exact MinNE/MaxNE is NP-hard, and only several heuristic algorithms are proposed to approximate the MinNE/MaxNE. To calculate the exact MinNE/MaxNE, we provide a thorough analysis of some special graphs and propose scalable and exact algorithms for general graphs. Especially, our contributions are four-fold. First, we analytically give the MinNE/MaxNE for EC on special graphs based on spectral radius. Second, we provide an integer linear programming formulation (ILP) to determine MinNE/MaxNE for the general graphs with the small epidemic threshold. Third, we propose a branch-and-bound (BnB) framework to compute the exact MinNE/MaxNE in the general graphs with several heuristic methods to branch the variables. Fourth, we adopt NetShiled (NetS) method to approximate the MinNE to improve the scalability. Extensive experiments demonstrate that our BnB algorithm can outperform the naive enumeration method in scalability, and the NetS can improve the scalability significantly and outperform the previous heuristic method in solution quality.

scholarly journals The average tree solution for cycle-free graph games

2008 ◽  
Vol 62 (1) ◽  
pp. 77-92 ◽  
Author(s):  
P. Jean Jacques Herings ◽  
Gerard van der Laan ◽  
Dolf Talman
Keyword(s):  
Free Graph ◽  
Graph Games

Every 3-connected {K 1,3,N 3,3,3}-free graph is Hamiltonian

2013 ◽  
Vol 56 (8) ◽  
pp. 1585-1595 ◽  
Author(s):  
HouYuan Lin ◽  
ZhiQuan Hu
Keyword(s):  
Free Graph
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