scholarly journals Remarks on the Local Irregularity Conjecture

Mathematics ◽  
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.

scholarly journals On Graphs with Cyclic Defect or Excess

10.37236/415 ◽  
2010 ◽  
Vol 17 (1) ◽  
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.

scholarly journals Disjoint Cycles of Different Lengths in Graphs and Digraphs

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

2002 ◽  
Vol 11 (1) ◽  
pp. 97-102 ◽  
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.

scholarly journals On the Strong Chromatic Index of Sparse Graphs

10.37236/5390 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Philip DeOrsey ◽  
Michael Ferrara ◽  
Nathan Graber ◽  
Stephen G. Hartke ◽  
Luke L. Nelsen ◽  
...  
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Maximum Degree ◽  
Applied Mathematics ◽  
Edge Coloring ◽  
Combinatorial Nullstellensatz ◽  
Chromatic Index ◽  
Sparse Graphs ◽  
Strong Chromatic Index ◽  
General Bound

The strong chromatic index of a graph $G$, denoted $\chi'_s(G)$, is the least number of colors needed to edge-color $G$ so that edges at distance at most two receive distinct colors. The strong list chromatic index, denoted $\chi'_{s,\ell}(G)$, is the least integer $k$ such that if arbitrary lists of size $k$ are assigned to each edge then $G$ can be edge-colored from those lists where edges at distance at most two receive distinct colors.We use the discharging method, the Combinatorial Nullstellensatz, and computation to show that if $G$ is a subcubic planar graph with ${\rm girth}(G) \geq 41$ then $\chi'_{s,\ell}(G) \leq 5$, answering a question of Borodin and Ivanova [Precise upper bound for the strong edge chromatic number of sparse planar graphs, Discuss. Math. Graph Theory, 33(4), (2014) 759--770]. We further show that if $G$ is a subcubic planar graph and ${\rm girth}(G) \geq 30$, then $\chi_s'(G) \leq 5$, improving a bound from the same paper.Finally, if $G$ is a planar graph with maximum degree at most four and ${\rm girth}(G) \geq 28$, then $\chi'_s(G)N \leq 7$, improving a more general bound of Wang and Zhao from [Odd graphs and its applications to the strong edge coloring, Applied Mathematics and Computation, 325 (2018), 246-251] in this case.

scholarly journals Covering a Graph with Cycles of Length at Least 4

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.

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

10.37236/7239 ◽  
2019 ◽  
Vol 26 (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.  

scholarly journals A fast $$(2 + \frac{2}{7})$$-approximation algorithm for capacitated cycle covering

2021 ◽  
Author(s):  
Vera Traub ◽  
Thorben Tröbst
Keyword(s):  
Vehicle Routing Problem ◽  
Covering Problem ◽  
Post Processing ◽  
Routing Problem ◽  
Cycle Cover ◽  
Disjoint Cycles ◽  
Processing Step ◽  
Number Of Cycles ◽  
Vertex Disjoint ◽  
Capacitated Vehicle

AbstractWe consider the capacitated cycle covering problem: given an undirected, complete graph G with metric edge lengths and demands on the vertices, we want to cover the vertices with vertex-disjoint cycles, each serving a demand of at most one. The objective is to minimize a linear combination of the total length and the number of cycles. This problem is closely related to the capacitated vehicle routing problem (CVRP) and other cycle cover problems such as min-max cycle cover and bounded cycle cover. We show that a greedy algorithm followed by a post-processing step yields a $$(2 + \frac{2}{7})$$ ( 2 + 2 7 ) -approximation for this problem by comparing the solution to a polymatroid relaxation. We also show that the analysis of our algorithm is tight and provide a $$2 + \epsilon $$ 2 + ϵ lower bound for the relaxation.

scholarly journals On Directed Versions of the Hajnal–Szemerédi Theorem

2015 ◽  
Vol 24 (6) ◽  
pp. 873-928 ◽  
Author(s):  
ANDREW TREGLOWN
Keyword(s):  
Directed Graph ◽  
Minimum Degree ◽  
Directed Graphs ◽  
Perfect Matchings ◽  
Large Order ◽  
Vertex Disjoint ◽  
Removal Lemma

We say that a (di)graph G has a perfect H-packing if there exists a set of vertex-disjoint copies of H which cover all the vertices in G. The seminal Hajnal–Szemerédi theorem characterizes the minimum degree that ensures a graph G contains a perfect Kr-packing. In this paper we prove the following analogue for directed graphs: Suppose that T is a tournament on r vertices and G is a digraph of sufficiently large order n where r divides n. If G has minimum in- and outdegree at least (1−1/r)n then G contains a perfect T-packing.In the case when T is a cyclic triangle, this result verifies a recent conjecture of Czygrinow, Kierstead and Molla [4] (for large digraphs). Furthermore, in the case when T is transitive we conjecture that it suffices for every vertex in G to have sufficiently large indegree or outdegree. We prove this conjecture for transitive triangles and asymptotically for all r ⩾ 3. Our approach makes use of a result of Keevash and Mycroft [10] concerning almost perfect matchings in hypergraphs as well as the Directed Graph Removal Lemma [1, 6].

scholarly journals A New Bound on the Domination Number of Graphs with Minimum Degree Two

10.37236/499 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Michael A. Henning ◽  
Ingo Schiermeyer ◽  
Anders Yeo
Keyword(s):  
Connected Graph ◽  
Minimum Degree ◽  
Domination Number ◽  
Degree Condition ◽  
Cut Vertex ◽  
Delta G ◽  
Vertex Disjoint ◽  
Special Cycle

For a graph $G$, let $\gamma(G)$ denote the domination number of $G$ and let $\delta(G)$ denote the minimum degree among the vertices of $G$. A vertex $x$ is called a bad-cut-vertex of $G$ if $G-x$ contains a component, $C_x$, which is an induced $4$-cycle and $x$ is adjacent to at least one but at most three vertices on $C_x$. A cycle $C$ is called a special-cycle if $C$ is a $5$-cycle in $G$ such that if $u$ and $v$ are consecutive vertices on $C$, then at least one of $u$ and $v$ has degree $2$ in $G$. We let ${\rm bc}(G)$ denote the number of bad-cut-vertices in $G$, and ${\rm sc}(G)$ the maximum number of vertex disjoint special-cycles in $G$ that contain no bad-cut-vertices. We say that a graph is $(C_4,C_5)$-free if it has no induced $4$-cycle or $5$-cycle. Bruce Reed [Paths, stars and the number three. Combin. Probab. Comput. 5 (1996), 277–295] showed that if $G$ is a graph of order $n$ with $\delta(G) \ge 3$, then $\gamma(G) \le 3n/8$. In this paper, we relax the minimum degree condition from three to two. Let $G$ be a connected graph of order $n \ge 14$ with $\delta(G) \ge 2$. As an application of Reed's result, we show that $\gamma(G) \le \frac{1}{8} ( 3n + {\rm sc}(G) + {\rm bc}(G))$. As a consequence of this result, we have that (i) $\gamma(G) \le 2n/5$; (ii) if $G$ contains no special-cycle and no bad-cut-vertex, then $\gamma(G) \le 3n/8$; (iii) if $G$ is $(C_4,C_5)$-free, then $\gamma(G) \le 3n/8$; (iv) if $G$ is $2$-connected and $d_G(u) + d_G(v) \ge 5$ for every two adjacent vertices $u$ and $v$, then $\gamma(G) \le 3n/8$. All bounds are sharp.

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