Infinite Family Of Cubic Graphs Research Articles

Locally irregular edge-coloring of subcubic graphs

A graph is locally irregular if no two adjacent vertices have the same degree. A locally irregular edge-coloring of a graph G is such an (improper) edge-coloring that the edges of any fixed color induce a locally irregular subgraph. Among the graphs admitting a locally irregular edge-coloring, i.e., decomposable graphs, only one is known to require 4 colors, while for all the others it is believed that 3 colors suffice. In this paper, we prove that decomposable claw-free graphs with maximum degree 3, all cycle permutation graphs, and all generalized Petersen graphs admit a locally irregular edge-coloring with at most 3 colors. We also discuss when 2 colors suffice for a locally irregular edge-coloring of cubic graphs and present an infinite family of cubic graphs of girth 4 which require 3 colors.

On 2-power unicyclic cubic graphs

In a graph, a cycle whose length is a power of two (that is, 2 k ) is called a 2-power cycle. In this paper, we show that the existence of an infinite family of cubic graphs which contain only one cycle whose length is a power of 2. Such graphs are called as 2-power unicyclic cubic graphs. Further we observe that the only 2-power cycle in a cubic graph cannot be removed implying that there does not exist a counter example for Erdos-Gyárfás conjecture.

Progress on fault-tolerant locating-dominating sets

A locating-dominating set is a subset of vertices representing “detectors” in a graph G which can locate an “intruder” given that each detector covers its closed neighborhood and can distinguish its own location from its neighbors. We explore a fault-tolerant variant of locating-dominating sets, called error-detecting locating-dominating (DET:LD) sets, which can tolerate one false negative. The concept of DET:LD sets was originally introduced in 2002, along with several properties in various classes of graphs; we extend this work by fully characterizing DET:LD sets for general graphs and establishing existence criteria. We prove that the problem of determining the minimum density of a DET:LD set in arbitrary graphs is NP-complete. Additionally, we determine that the minimum density in cubic graphs is at least [Formula: see text], and we show an infinite family of cubic graphs achieving this density.

Vertex-edge domination in cubic graphs

We establish that any connected cubic graph of order n>6 has a minimum vertex-edge dominating set of at most 10n31 vertices, thus affirmatively answering the open question posed by Klostermeyer et al. in Discussiones Mathematicae Graph Theory, https://doi.org/10.7151/dmgt.2175. On the other hand, we present an infinite family of cubic graphs whose γve ratio is equal to 27. Finally, we show that the problem of determining the minimum γve-dominating set is NP-hard even in cubic planar graphs.

Small Snarks with Large Oddness

We estimate the minimum number of vertices of a cubic graph with given oddness and cyclic connectivity. We prove that a bridgeless cubic graph $G$ with oddness $\omega(G)$ other than the Petersen graph has at least $5.41\, \omega(G)$ vertices, and for each integer $k$ with $2\le k\le 6$ we construct an infinite family of cubic graphs with cyclic connectivity $k$ and small oddness ratio $|V(G)|/\omega(G)$. In particular, for cyclic connectivity $2$, $4$, $5$, and $6$ we improve the upper bounds on the oddness ratio of snarks to $7.5$, $13$, $25$, and $99$ from the known values $9$, $15$, $76$, and $118$, respectively. In addition, we construct a cyclically $4$-connected snark of girth $5$ with oddness $4$ on $44$ vertices, improving the best previous value of $46$. Corrigendum added March 19, 2018.

Cubic Graphs with Given Circular Chromatic Index

For every rational number r such that $3<r<3+1/3$, we construct an infinite family of cubic graphs with circular chromatic index r. Our construction disproves a conjecture of Zhu that there is no bounded increasing infinite sequence in the set of circular chromatic indices of graphs [Topics in Discrete Mathematics, Dedicated to Jarik Neetil on the Occasion of his 60th Birthday, Algorithms Combin. 26, M. Klazar, J. Kratochvíl, M. Loebl, J. Matoušek, R. Thomas, and P. Valtr, eds., Springer, Berlin, 2006, pp. 497–550]. We also describe a sequence of graphs whose circular chromatic indices converge to $3+1/3$ from above.

A note on domination and minus domination numbers in cubic graphs

Let G = ( V , E ) be a graph. A subset S of V is called a dominating set if each vertex of V − S has at least one neighbor in S . The domination number γ ( G ) equals the minimum cardinality of a dominating set in G . A minus dominating function on G is a function f : V → { − 1 , 0 , 1 } such that f ( N [ v ] ) = ∑ u ∈ N [ v ] f ( u ) ≥ 1 for each v ∈ V , where N [ v ] is the closed neighborhood of v . The minus domination number of G is γ − ( G ) = min { ∑ v ∈ V f ( v ) ∣ f is a minus dominating function on G } . It was incorrectly shown in [X. Yang, Q. Hou, X. Huang, H. Xuan, The difference between the domination number and minus domination number of a cubic graph, Applied Mathematics Letters 16 (2003) 1089–1093] that there is an infinite family of cubic graphs in which the difference γ − γ − can be made arbitrary large. This note corrects the mistakes in the proof and poses a new problem on the upper bound for γ − γ − in cubic graphs.

The difference between the domination number and the minus domination number of a cubic graph

The closed neighborhood of a vertex subset S of a graph G = ( V, E), denoted as N[ S], is defined as the union of S and the set of all the vertices adjacent to some vertex of S. A dominating set of a graph G = ( V, E) is defined as a set S of vertices such that N[ S] = V. The domination number of a graph G, denoted as γ( G), is the minimum possible size of a dominating set of G. A minus dominating function on a graph G = ( V, E) is a function g : V → {−1, 0, 1} such that g( N[ v]) ≥ 1 for all vertices. The weight of a minus dominating function g is defined as g( V) = Σ vϵV g( v) . The minus domination number of a graph G, denoted as γ −( G), is the minimum possible weight of a minus dominating function on G. It is well known that γ −( G) ≤ γ( G). This paper is focused on the difference between γ( G) and γ −( G) for cubic graphs. We first present a graph-theoretic description of γ −( G). Based on this, we give a necessary and sufficient condition for γ( G) − γ −( G) ≥ k. Further, we present an infinite family of cubic graphs of order 18 k + 16 and with γ( G) − γ −( G) ≥ k