Filters








35,027 Hits in 6.2 sec

Cubic graphs with equal independence number and matching number [article]

Elena Mohr, Dieter Rautenbach
2019 arXiv   pre-print
Caro, Davila, and Pepper (arXiv:1909.09093) recently proved δ(G) α(G)≤Δ(G) μ(G) for every graph G with minimum degree δ(G), maximum degree Δ(G), independence number α(G), and matching number μ(G).  ...  Answering some problems they posed, we characterize the extremal graphs for δ(G)<Δ(G) as well as for δ(G)=Δ(G)=3.  ...  Acknowledgment We thank Yair Caro, Randy Davila, and Ryan Pepper for valuable discussion and for sharing their many crucial examples of extremal graphs.  ... 
arXiv:1910.11762v1 fatcat:zoepwm2yxbforov2i3654dw6lq

On the maximum number of independent edges in cubic graphs

Arthur M. Hobbs, Edward Schmeichel
1982 Discrete Mathematics  
Given an extended real number r, a property P(r) of graphs is super-hereditary if, whenever graph G has property P(r) and H is a subgraph of G, then H has property P(s) with s I> r.  ...  Let the reals be extended to include oo with o~ > r for every real nuraber r.  ...  is a cubic graph with n vertices and property P, then G contains a matching with at least ½n(3g-1)/(3g-~-1) edges.  ... 
doi:10.1016/0012-365x(82)90227-8 fatcat:qbuy5aaecnfsllzyo33maeklzu

Page 1494 of Mathematical Reviews Vol. , Issue 2001C [page]

2001 Mathematical Reviews  
We provide a constructive characterisation of those trees with equal domination and restrained domination numbers.  ...  Summary: “It is known that finding a perfect matching in a general graph is AC’-equivalent to finding a perfect matching in a 3-regular (i.e. cubic) graph.  ... 

Matchings and Nonrainbow Colorings

Zdeněk Dvořák, Stanislav Jendrol', Daniel Král', Gyula Pap
2009 SIAM Journal on Discrete Mathematics  
We show that the maximum number of colors that can be used in a vertex coloring of a cubic 3-connected plane graph G that avoids a face with vertices of mutually distinct colors (a rainbow face) is equal  ...  to n 2 + µ * − 2, where n is the number of vertices of G and µ * is the size of the maximum matching of the dual graph G * .  ...  In particular, n 2 + μ * − 2 ≤ χ f (G) ≤ n − α * for connected cubic plane graphs G [10] , where α * is the independence number of the dual graph G * of G and μ * is the size of the largest matching of  ... 
doi:10.1137/060675927 fatcat:oijob43zdfgvja5h23cg7anu2i

Average connectivity and average edge-connectivity in graphs

Jaehoon Kim, Suil O
2013 Discrete Mathematics  
In this paper, we prove a relationship between the average connectivity and the matching number in all graphs.  ...  We also give the best lower bound for the average edge-connectivity over n-vertex connected cubic graphs, and we characterize the graphs where equality holds.  ...  We will show that equality holds only graphs in a family. The family is also useful to find the minimum number of perfect matchings over n-vertex cubic graphs having a perfect matching.  ... 
doi:10.1016/j.disc.2013.05.024 fatcat:watglpdtynbprf3fdrnjqdvvgy

CLIQUE-TRANSVERSAL SETS IN LINE GRAPHS OF CUBIC GRAPHS AND TRIANGLE-FREE GRAPHS

LIYING KANG, ERFANG SHAN
2015 Bulletin of the Korean Mathematical Society  
For every cubic graph with at most two bridges, we first show that it has a perfect matching which contains exactly one edge of each triangle of it; by the result, we determine the exact value of the clique-transversal  ...  Also, we present a sharp upper bound on the cliquetransversal number of line graph of a cubic graph.  ...  If G is a cubic graph of order n with girth r ≥ 3, then α ′ (G) ≥ 3g−1 3g+1 n 2 , where g is the number of vertices in B. Moreover, the equality holds if and only if G is the graph H(g). holds.  ... 
doi:10.4134/bkms.2015.52.5.1423 fatcat:bcp67yza4ncmlcluvdopk5sscm

Zero Forcing in Claw-Free Cubic Graphs [article]

Randy Davila, Michael Henning
2018 arXiv   pre-print
In this paper we show a surprising relation between the zero forcing number of a graph and the independence number of a graph, denoted α(G).  ...  As a consequence of this result, if G K_4 is a connected, cubic, claw-free graph with order n, then Z(G) <2/5n + 1.  ...  First to establish a relationship between the zero forcing number of a cubic, claw-free graph and its independence and matching numbers.  ... 
arXiv:1802.03108v1 fatcat:vyrdfeghtrecbftggmnmelopfu

Domination number of cubic graphs with large girth [article]

Daniel Kral, Petr Skoda, Jan Volec
2009 arXiv   pre-print
We show that every n-vertex cubic graph with girth at least g have domination number at most 0.299871n+O(n/g)<3n/10+O(n/g).  ...  The authors would like to thank RisteŠkrekovski who was hosting them for an extra-ordinary working conditions and for numerous discussions on the subject.  ...  Fix a cubic bridgeless graph G with girth at least g and also fix an integer K which determines the number of levels as defined later.  ... 
arXiv:0907.1166v1 fatcat:seycafb2ifcydbnyzc7qhldmee

Subgroup sum graphs of finite abelian groups [article]

Peter J. Cameron, R. Raveendra Prathap, T. Tamizh Chelvam
2021 arXiv   pre-print
We study perfectness, clique number and independence number, connectedness, diameter, spectrum, and domination number of these graphs and their complements.  ...  The subgroup sum graph Γ_G,H is the graph with vertex set G, in which two distinct vertices x and y are joined if x+y∈ H∖{0}. These graphs form a fairly large class of Cayley sum graphs.  ...  Tamizh Chelvam is supported by CSIR Emeritus Scientist Scheme (No.21 (1123)/20/EMR-II) of Council of Scientific and Industrial Research, Government of India.  ... 
arXiv:2111.05748v1 fatcat:xlfssinwdjaq7e5kjoo32fapie

More on the sixth coefficient of the matching polynomial in regular graphs [article]

Neda Soltani, Saeid Alikhani
2017 arXiv   pre-print
We find the sixth coefficient of the matching polynomial of regular graphs. As a consequence, every cubic graph of order 10 is matching unique.  ...  A matching set M in a graph G is a collection of edges of G such that no two edges from M share a vertex. In this paper we consider some parameters related to the matching of regular graphs.  ...  In the next section, after investigation of the saturation number and the matching number of cubic graphs of order 10, we establish a formula for the number of 5-matchings in regular graphs.  ... 
arXiv:1710.07426v1 fatcat:pshlnxywwjdhtpubbo7s7kjyoy

Domination number of cubic graphs with large girth

Daniel Král', Petr Škoda, Jan Volec
2011 Journal of Graph Theory  
We show that every n-vertex cubic graph with girth at least g have domination number at most 0.299871n + O (n/g) < 3n/10 + O (n/g).  ...  The authors would like to thank RisteŠkrekovski who was hosting them for an extra-ordinary working conditions and for numerous discussions on the subject.  ...  [2] for bridgeless n-vertex cubic graphs with girth at least g for g divisible by three and 1 3 + 8 3g 2 n of Kostochka and Stodolsky [5] for all n-vertex cubic graphs with girth at least g.  ... 
doi:10.1002/jgt.20568 fatcat:js2s7iiwhrfvhbtuflk4dgyykq

Domination in Cubic Graphs of Large Girth [chapter]

Dieter Rautenbach, Bruce Reed
2008 Lecture Notes in Computer Science  
We prove that connected cubic graphs of order n and girth g have domination number at most 0.32127n + O n g .  ...  Since for fixed d and g the numbers of cycles in random cubic graphs of fixed lengths r < g are asymptotically distributed as independent Poisson variables [2] with mean (d − 1) r /2r, the number of  ...  Let H denote the graph with vertex set V \ D 0 whose edges are the edges of the paths in P together with the matching M . Note that M is a perfect matching of H.  ... 
doi:10.1007/978-3-540-89550-3_20 fatcat:5p3djeq36vdfjfg4k5pxfocg4m

Perfect matchings of line graphs with small maximum degree [article]

Weigen Yan, Fuji Zhang
2009 arXiv   pre-print
As a corollary, we prove that the number of perfect matchings of a connected cubic line graph with n vertices equals 2^n/6+1 if n>4, which implies the conjecture by Lovász and Plummer holds for the connected  ...  We show that if G has an even number of edges, then the number of perfect matchings of the line graph of G equals 2^n/2+1, where n is the number of 3-degree vertices of G.  ...  As a corollary, we also prove that the number of perfect matchings of a connected cubic line graph with n vertices equals 2 n/6+1 if n > 4.  ... 
arXiv:0906.3873v1 fatcat:muf5uellzbgxxe4kfbbb7g7nv4

Largest 2-regular subgraphs in 3-regular graphs [article]

Ilkyoo Choi, Ringi Kim, Alexandr Kostochka, Boram Park, Douglas B. West
2019 arXiv   pre-print
For a graph G, let f_2(G) denote the largest number of vertices in a 2-regular subgraph of G. We determine the minimum of f_2(G) over 3-regular n-vertex simple graphs G.  ...  More generally, every n-vertex multigraph with maximum degree 3 and m edges has a 2-regular subgraph that omits at most {0, (3n-2m+c-1)/2} vertices.  ...  Since O and West [9] showed that a cubic n-vertex graph has at most (n − 7)/3 cut-edges, Theorem 1.1 immediately yields a lower bound on f 2 (G) for a cubic graph G in terms of the number of vertices  ... 
arXiv:1903.08795v1 fatcat:bciesaaemrcl5h3gdyost7p6t4

Minimum maximal matchings in cubic graphs [article]

Wouter Cames van Batenburg
2021 arXiv   pre-print
We prove that every connected cubic graph with n vertices has a maximal matching of size at most 5/12 n+ 1/2.  ...  More generally, we prove that every graph with n vertices and m edges and maximum degree at most 3 has a maximal matching of size at most 4n-m/6+ 1/2.  ...  The author thanks the two anonymous reviewers for their detailed comments and suggestions.  ... 
arXiv:2008.01863v3 fatcat:o6t5qszo3fakpedk3dwkaolfi4
« Previous Showing results 1 — 15 out of 35,027 results