On Graphs with Cyclic Defect or Excess

Charles Delorme; Guillermo Pineda-Villavicencio

doi:10.37236/415

On Graphs with Cyclic Defect or Excess

The Electronic Journal of Combinatorics ◽

10.37236/415 ◽

2010 ◽

Vol 17 (1) ◽

Cited By ~ 3

Author(s):

Charles Delorme ◽

Guillermo Pineda-Villavicencio

Keyword(s):

Upper Bound ◽

Adjacency Matrix ◽

Minimum Degree ◽

Integer Coefficients ◽

Disjoint Cycles ◽

The Matrix ◽

Odd Degree ◽

Unexplored Area ◽

Trivial Graph ◽

Vertex Disjoint

The Moore bound constitutes both an upper bound on the order of a graph of maximum degree $d$ and diameter $D=k$ and a lower bound on the order of a graph of minimum degree $d$ and odd girth $g=2k+1$. Graphs missing or exceeding the Moore bound by $\epsilon$ are called graphs with defect or excess $\epsilon$, respectively. While Moore graphs (graphs with $\epsilon=0$) and graphs with defect or excess 1 have been characterized almost completely, graphs with defect or excess 2 represent a wide unexplored area. Graphs with defect (excess) 2 satisfy the equation $G_{d,k}(A) = J_n + B$ ($G_{d,k}(A) = J_n - B$), where $A$ denotes the adjacency matrix of the graph in question, $n$ its order, $J_n$ the $n\times n$ matrix whose entries are all 1's, $B$ the adjacency matrix of a union of vertex-disjoint cycles, and $G_{d,k}(x)$ a polynomial with integer coefficients such that the matrix $G_{d,k}(A)$ gives the number of paths of length at most $k$ joining each pair of vertices in the graph. In particular, if $B$ is the adjacency matrix of a cycle of order $n$ we call the corresponding graphs graphs with cyclic defect or excess; these graphs are the subject of our attention in this paper. We prove the non-existence of infinitely many such graphs. As the highlight of the paper we provide the asymptotic upper bound of $O(\frac{64}3d^{3/2})$ for the number of graphs of odd degree $d\ge3$ and cyclic defect or excess. This bound is in fact quite generous, and as a way of illustration, we show the non-existence of some families of graphs of odd degree $d\ge3$ and cyclic defect or excess. Actually, we conjecture that, apart from the Möbius ladder on 8 vertices, no non-trivial graph of any degree $\ge 3$ and cyclic defect or excess exists.

Graphs with many vertex-disjoint cycles

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.587 ◽

2012 ◽

Vol Vol. 14 no. 2 (Graph Theory) ◽

Author(s):

Dieter Rautenbach ◽

Friedrich Regen

Keyword(s):

Graph Theory ◽

Upper Bound ◽

Linear Time ◽

Simple Extension ◽

Disjoint Cycles ◽

Cyclomatic Number ◽

Finite Set ◽

Extension Rule ◽

International Audience ◽

Vertex Disjoint

Graph Theory International audience We study graphs G in which the maximum number of vertex-disjoint cycles nu(G) is close to the cyclomatic number mu(G), which is a natural upper bound for nu(G). Our main result is the existence of a finite set P(k) of graphs for all k is an element of N-0 such that every 2-connected graph G with mu(G)-nu(G) = k arises by applying a simple extension rule to a graph in P(k). As an algorithmic consequence we describe algorithms calculating minmu(G)-nu(G), k + 1 in linear time for fixed k.

New Bounds for the α-Indices of Graphs

Mathematics ◽

10.3390/math8101668 ◽

2020 ◽

Vol 8 (10) ◽

pp. 1668

Author(s):

Eber Lenes ◽

Exequiel Mallea-Zepeda ◽

Jonnathan Rodríguez

Keyword(s):

Lower Bound ◽

Spectral Radius ◽

Upper Bound ◽

Adjacency Matrix ◽

Independence Number ◽

Minimal Degree ◽

Diagonal Matrix ◽

The Matrix

Let G be a graph, for any real 0≤α≤1, Nikiforov defines the matrix Aα(G) as Aα(G)=αD(G)+(1−α)A(G), where A(G) and D(G) are the adjacency matrix and diagonal matrix of degrees of the vertices of G. This paper presents some extremal results about the spectral radius ρα(G) of the matrix Aα(G). In particular, we give a lower bound on the spectral radius ρα(G) in terms of order and independence number. In addition, we obtain an upper bound for the spectral radius ρα(G) in terms of order and minimal degree. Furthermore, for n>l>0 and 1≤p≤⌊n−l2⌋, let Gp≅Kl∨(Kp∪Kn−p−l) be the graph obtained from the graphs Kl and Kp∪Kn−p−l and edges connecting each vertex of Kl with every vertex of Kp∪Kn−p−l. We prove that ρα(Gp+1)<ρα(Gp) for 1≤p≤⌊n−l2⌋−1.

On Komlós’ tiling theorem in random graphs

Combinatorics Probability Computing ◽

10.1017/s0963548319000129 ◽

2019 ◽

Vol 29 (1) ◽

pp. 113-127

Author(s):

Rajko Nenadov ◽

Nemanja Škorić

Keyword(s):

Chromatic Number ◽

Random Graph ◽

Random Graphs ◽

Upper Bound ◽

Minimum Degree ◽

Arbitrary Graph ◽

Spanning Subgraph ◽

Delta G ◽

Image Position ◽

Vertex Disjoint

AbstractGiven graphs G and H, a family of vertex-disjoint copies of H in G is called an H-tiling. Conlon, Gowers, Samotij and Schacht showed that for a given graph H and a constant γ>0, there exists C>0 such that if $p \ge C{n^{ - 1/{m_2}(H)}}$ , then asymptotically almost surely every spanning subgraph G of the random graph 𝒢(n, p) with minimum degree at least $\delta (G) \ge (1 - \frac{1}{{{\chi _{{\rm{cr}}}}(H)}} + \gamma )np$ contains an H-tiling that covers all but at most γn vertices. Here, χcr(H) denotes the critical chromatic number, a parameter introduced by Komlós, and m2(H) is the 2-density of H. We show that this theorem can be bootstrapped to obtain an H-tiling covering all but at most $\gamma {(C/p)^{{m_2}(H)}}$ vertices, which is strictly smaller when $p \ge C{n^{ - 1/{m_2}(H)}}$ . In the case where H = K3, this answers the question of Balogh, Lee and Samotij. Furthermore, for an arbitrary graph H we give an upper bound on p for which some leftover is unavoidable and a bound on the size of a largest H -tiling for p below this value.

Disjoint Cycles of Different Lengths in Graphs and Digraphs

The Electronic Journal of Combinatorics ◽

10.37236/6921 ◽

2017 ◽

Vol 24 (4) ◽

Author(s):

Julien Bensmail ◽

Ararat Harutyunyan ◽

Ngoc Khang Le ◽

Binlong Li ◽

Nicolas Lichiardopol

Keyword(s):

Minimum Degree ◽

Directed Graphs ◽

Probabilistic Method ◽

Undirected Graphs ◽

Disjoint Cycles ◽

Vertex Disjoint ◽

Degree Bound ◽

Directed Cycles

In this paper, we study the question of finding a set of $k$ vertex-disjoint cycles (resp. directed cycles) of distinct lengths in a given graph (resp. digraph). In the context of undirected graphs, we prove that, for every $k \geq 1$, every graph with minimum degree at least $\frac{k^2+5k-2}{2}$ has $k$ vertex-disjoint cycles of different lengths, where the degree bound is best possible. We also consider other cases such as when the graph is triangle-free, or the $k$ cycles are required to have different lengths modulo some value $r$. In the context of directed graphs, we consider a conjecture of Lichiardopol concerning the least minimum out-degree required for a digraph to have $k$ vertex-disjoint directed cycles of different lengths. We verify this conjecture for tournaments, and, by using the probabilistic method, for some regular digraphs and digraphs of small order.

A Note on Vertex-Disjoint Cycles

Combinatorics Probability Computing ◽

10.1017/s0963548301004904 ◽

2002 ◽

Vol 11 (1) ◽

pp. 97-102 ◽

Cited By ~ 3

Author(s):

JACQUES VERSTRAËTE

Keyword(s):

Minimum Degree ◽

Quadratic Function ◽

Average Degree ◽

Disjoint Cycles ◽

Vertex Disjoint

Häggkvist and Scott asked whether one can find a quadratic function q(k) such that, if G is a graph of minimum degree at least q(k), then G contains vertex-disjoint cycles of k consecutive even lengths. In this paper, it is shown that if G is a graph of average degree at least k2+19k+10 with sufficiently many vertices, then G contains vertex-disjoint cycles of k consecutive even lengths, answering the above question in the affirmative. The coefficient of k2 cannot be decreased and, in this sense, this result is best possible.

Edge-Maximal Graphs Containing No $r$ Vertex-Disjoint Triangles

Journal of Mathematics Research ◽

10.5539/jmr.v10n1p110 ◽

2017 ◽

Vol 10 (1) ◽

pp. 110

Author(s):

Mohammad Hailat

Keyword(s):

Graph Theory ◽

Upper Bound ◽

Tripartite Graph ◽

Disjoint Cycles ◽

Large N ◽

Vertex Disjoint

An important problem in graph theory is that of determining the maximum number of edges in a given graph $G$ that contains no specific subgraphs. This problem has attracted the attention of many researchers. An example of such a problem is the determination of an upper bound on the number of edges of a graph that has no triangles. In this paper, we let $\mathcal{G}(n,V_{r,3})$ denote the class of graphs on $n$ vertices containing no $r$-vertex-disjoint cycles of length $3$. We show that for large $n$, $\mathcal{E}(G)\les \lfloor \frac{(n-r+1)^2}{4} \rfloor +(r-1)(n-r+1)$ for every $G\in\mathcal{G}(n,V_{r,3})$. Furthermore, equality holds if and only if $G=\Omega(n,r)=K_{r-1,\lfloor \frac{n-r+1}2\rfloor,\lceil \frac{n-r+1}2\rceil}$ where $\Omega(n,r)$ is a tripartite graph on $n$ vertices.

Covering a Graph with Cycles of Length at Least 4

The Electronic Journal of Combinatorics ◽

10.37236/4099 ◽

2018 ◽

Vol 25 (1) ◽

Author(s):

Hong Wang

Keyword(s):

Positive Integer ◽

Minimum Degree ◽

Disjoint Cycles ◽

Vertex Disjoint

Let $G$ be a graph of order $n\geq 4k$, where $k$ is a positive integer. Suppose that the minimum degree of $G$ is at least $\lceil n/2\rceil$. We show that $G$ contains $k$ vertex-disjoint cycles covering all the vertices of $G$ such that $k-1$ of them are quadrilaterals.

On Bipartite Cages of Excess 4

The Electronic Journal of Combinatorics ◽

10.37236/6693 ◽

2017 ◽

Vol 24 (1) ◽

Cited By ~ 1

Author(s):

Slobodan Filipovski

Keyword(s):

Adjacency Matrix ◽

Disjoint Union ◽

Necessary Conditions ◽

Distance Matrix ◽

Bipartite Graphs ◽

Regular Graphs ◽

Dickson Polynomial ◽

Disjoint Cycles ◽

The Difference ◽

Vertex Disjoint

The Moore bound $M(k,g)$ is a lower bound on the order of $k$-regular graphs of girth $g$ (denoted $(k,g)$-graphs). The excess $e$ of a $(k,g)$-graph of order $n$ is the difference $ n-M(k,g) $. In this paper we consider the existence of $(k,g)$-bipartite graphs of excess $4$ by studying spectral properties of their adjacency matrices. For a given graph $G$ and for the integers $i$ with $0\leq i\leq diam(G)$, the $i$-distance matrix $A_i$ of $G$ is an $n\times n$ matrix such that the entry in position $(u,v)$ is $1$ if the distance between the vertices $u$ and $v$ is $i$, and zero otherwise. We prove that the $(k,g)$-bipartite graphs of excess $4$ satisfy the equation $kJ=(A+kI)(H_{d-1}(A)+E)$, where $A=A_{1}$ denotes the adjacency matrix of the graph in question, $J$ the $n \times n$ all-ones matrix, $E=A_{d+1}$ the adjacency matrix of a union of vertex-disjoint cycles, and $H_{d-1}(x)$ is the Dickson polynomial of the second kind with parameter $k-1$ and degree $d-1$. We observe that the eigenvalues other than $\pm k$ of these graphs are roots of the polynomials $H_{d-1}(x)+\lambda$, where $\lambda$ is an eigenvalue of $E$. Based on the irreducibility of $H_{d-1}(x)\pm2$, we give necessary conditions for the existence of these graphs. If $E$ is the adjacency matrix of a cycle of order $n$, we call the corresponding graphs graphs with cyclic excess; if $E$ is the adjacency matrix of a disjoint union of two cycles, we call the corresponding graphs graphs with bicyclic excess. In this paper we prove the non-existence of $(k,g)$-graphs with cyclic excess $4$ if $k\geq6$ and $k \equiv1 \!\! \pmod {3}$, $g=8, 12, 16$ or $k \equiv2 \!\! \pmod {3}$, $g=8;$ and the non-existence of $(k,g)$-graphs with bicyclic excess $4$ if $k\geq7$ is an odd number and $g=2d$ such that $d\geq4$ is even.

Monochromatic Cycle Partitions of $2$-Coloured Graphs with Minimum Degree $3n/4$

The Electronic Journal of Combinatorics ◽

10.37236/7239 ◽

2019 ◽

Vol 26 (1) ◽

Cited By ~ 1

Author(s):

Shoham Letzter

Keyword(s):

Minimum Degree ◽

Disjoint Cycles ◽

Large N ◽

Vertex Set ◽

Vertex Disjoint ◽

Edge Colouring

Balogh, Barát, Gerbner, Gyárfás, and Sárközy made the following conjecture. Let $G$ be a graph on $n$ vertices with minimum degree at least $3n/4$. Then for every $2$-edge-colouring of $G$, the vertex set $V(G)$ may be partitioned into two vertex-disjoint cycles, one of each colour. We prove this conjecture for large $n$, improving approximate results by the aforementioned authors and by DeBiasio and Nelsen.

Remarks on the Local Irregularity Conjecture

Mathematics ◽

10.3390/math9243209 ◽

2021 ◽

Vol 9 (24) ◽

pp. 3209

Author(s):

Jelena Sedlar ◽

Riste Škrekovski

Keyword(s):

Minimum Degree ◽

Edge Coloring ◽

Chromatic Index ◽

Sparse Graphs ◽

Unicyclic Graphs ◽

Disjoint Cycles ◽

Cactus Graph ◽

Colorable Graph ◽

Vertex Disjoint ◽

Local Irregularity

A locally irregular graph is a graph in which the end vertices of every edge have distinct degrees. A locally irregular edge coloring of a graph G is any edge coloring of G such that each of the colors induces a locally irregular subgraph of G. A graph G is colorable if it allows a locally irregular edge coloring. The locally irregular chromatic index of a colorable graph G, denoted by χirr′(G), is the smallest number of colors used by a locally irregular edge coloring of G. The local irregularity conjecture claims that all graphs, except odd-length paths, odd-length cycles and a certain class of cacti are colorable by three colors. As the conjecture is valid for graphs with a large minimum degree and all non-colorable graphs are vertex disjoint cacti, we study rather sparse graphs. In this paper, we give a cactus graph B which contradicts this conjecture, i.e., χirr′(B)=4. Nevertheless, we show that the conjecture holds for unicyclic graphs and cacti with vertex disjoint cycles.