Cubic Graph Of Order Research Articles

PBIB-Designs Arising From BI-Connected Dominating Sets of Cubic Graphs of Order at Most 12

A $(v, b, \gamma_{bc}, r, \lambda_{i})$-design over regular graph $G = (V, E)$ of degree $k$ is an ordered pair $D = (V, B)$, where $|V| = v$ and $B$ is the set of bi-connected dominating sets of $G$ called blocks such that two vertices $\alpha$ and $\beta$ which are $i^{th}$ associates occur together in $ \lambda_{i}$ blocks, the numbers $\lambda_{i}$ being independent of the choice of the pair $\alpha$ and $\beta$. In this paper, we obtain Partially Balanced Incomplete Block (PBIB)-designs arising from bi-connected dominating sets in cubic graphs. Also, we give a complete list of PBIB-designs with respect to the bi-connected dominating sets for cubic graphs of order at most $12$. The discussion of non-existence of some designs corresponding to bi-connected dominating sets from certain graphs concludes the article.

Best possible upper bounds on the restrained domination number of cubic graphs

AbstractA dominating set in a graph is a set of vertices such that every vertex in is adjacent to a vertex in . A restrained dominating set of is a dominating set with the additional restraint that the graph obtained by removing all vertices in is isolate‐free. The domination number and the restrained domination number are the minimum cardinalities of a dominating set and restrained dominating set, respectively, of . Let be a cubic graph of order . A classical result of Reed states that , and this bound is best possible. To determine the best possible upper bound on the restrained domination number of is more challenging, and we prove that .

Decycling cubic graphs

A set of vertices of a graph G is said to be decycling if its removal leaves an acyclic subgraph. The size of a smallest decycling set is the decycling number of G. Generally, at least ⌈(n+2)/4⌉ vertices have to be removed in order to decycle a cubic graph on n vertices. In 1979, Payan and Sakarovitch proved that the decycling number of a cyclically 4-edge-connected cubic graph of order n equals ⌈(n+2)/4⌉. In addition, they characterised the structure of minimum decycling sets and their complements. If n≡2(mod4), then G has a decycling set which is independent and its complement induces a tree. If n≡0(mod4), then one of two possibilities occurs: either G has an independent decycling set whose complement induces a forest of two trees, or the decycling set is near-independent (which means that it induces a single edge) and its complement induces a tree. In this paper we strengthen the result of Payan and Sakarovitch by proving that the latter possibility (a near-independent set and a tree) can always be guaranteed. Moreover, we relax the assumption of cyclic 4-edge-connectivity to a significantly weaker condition expressed through the canonical decomposition of 3-connected cubic graphs into cyclically 4-edge-connected ones. Our methods substantially use a surprising and seemingly distant relationship between the decycling number and the maximum genus of a cubic graph.

Partial domination in supercubic graphs

For some α with 0<α≤1, a subset X of vertices in a graph G of order n is an α-partial dominating set of G if the set X dominates at least α×n vertices in G. The α-partial domination number pdα(G) of G is the minimum cardinality of an α-partial dominating set of G. In this paper partial domination of graphs with minimum degree at least 3 is studied. It is proved that if G is a graph of order n and with δ(G)≥3, then pd78(G)≤13n. If in addition n≥60, then pd910(G)≤13n, and if G is a connected cubic graph of order n≥28, then pd1314(G)≤13n. Along the way it is shown that there are exactly four connected cubic graphs of order 14 with domination number 5.

A Note on Isolate Domination Number of a Cubic Graph

In this note we provide a solution to the problem “Find a structural characterization of cubic graph for which the isolate domination number equals one plus its domination number.” We show that if G is a cubic graph of order n and if 6 | n, then the isolate domination number of G is the same as the domination number of G. We also prove that if G is a connected cubic graph with diam(G) &gt; 2, then the isolate domination number is the same as the domination number.

Proper (Strong) Rainbow Connection and Proper (Strong) Rainbow Vertex Connection of Some Special Graphs

The proper rainbow vertex connection number of [Formula: see text], denoted by [Formula: see text], is the smallest number of colors needed to properly color the vertices of [Formula: see text] so that [Formula: see text] is rainbow vertex connected. The proper strong rainbow vertex connection number of [Formula: see text], denoted by [Formula: see text], is the smallest number of colors needed to properly color the vertices of [Formula: see text] so that [Formula: see text] is strong rainbow vertex connected. These two concepts are inspired by the concept of proper (strong) rainbow connection number of graphs. In this paper, we first determine the values of [Formula: see text] and [Formula: see text] for some special graphs, such as all cubic graphs of order [Formula: see text], pencil graphs, circular ladders or Möbius ladders. Secondly, we obtain the values of [Formula: see text] and [Formula: see text] for some special graphs, such as all cubic graphs of order [Formula: see text], paths, cycles, wheels, complete multipartite graphs, pencil graphs, circular ladders and Möbius ladders. Finally, we characterize all the connected graphs [Formula: see text] with [Formula: see text] and [Formula: see text].

The total vertex irregularity strength of symmetric cubic graphs of the Foster's Census

&lt;p&gt;Foster (1932) performed a mathematical census for all connected symmetric cubic (trivalent) graphs of order &lt;em&gt;n&lt;/em&gt; with &lt;em&gt;n &lt;/em&gt;≤ 512. This census then was continued by Conder et al. (2006) and they obtained the complete list of all connected symmetric cubic graphs with order &lt;em&gt;n&lt;/em&gt; ≤ 768. In this paper, we determine the total vertex irregularity strength of such graphs obtained by Foster. As a result, all the values of the total vertex irregularity strengths of the symmetric cubic graphs of order &lt;em&gt;n&lt;/em&gt; from Foster census strengthen the conjecture stated by Nurdin, Baskoro, Gaos &amp;amp; Salman (2010), namely ⌈(&lt;em&gt;n&lt;/em&gt;+3)/4⌉.&lt;/p&gt;

Perfect 3-colorings of Cubic Graphs of Order 8

Perfect coloring is a generalization of the notion of completely regular codes, given by Delsarte. A perfect $m$-coloring of a graph $G$ with $m$ colors is a partition of the vertex set of $G$ into m parts $A_1$, $\dots$, $A_m$ such that, for all $ i,j\in \lbrace 1,\cdots ,m\rbrace $, every vertex of $A_i$ is adjacent to the same number of vertices, namely, $a_{ij}$ vertices, of $A_j$ . The matrix $A=(a_{ij})_{i,j\in \lbrace 1,\cdots ,m\rbrace }$ is called the parameter matrix. We study the perfect 3-colorings (also known as the equitable partitions into three parts) of the cubic graphs of order $8$. In particular, we classify all the realizable parameter matrices of perfect 3-colorings for the cubic graphs of order 8.

Power domination in cubic graphs and Cartesian products

The power domination problem focuses on finding the optimal placement of phase measurement units (PMUs) to monitor an electrical power network. In the context of graphs, the power domination number of a graph G, denoted γP(G), is the minimum number of vertices needed to observe every vertex in the graph according to a specific set of observation rules. In [15], Zhao et al. proved that if G is a connected claw-free cubic graph of order n, then γP(G)≤n/4. In this paper, we show that if G is a claw-free diamond-free cubic graph of order n, then γP(G)≤n/6, and this bound is sharp. We also provide new bounds on γP(G□H) where G□H is the Cartesian product of graphs G and H. In the specific case that G and H are trees whose power domination number and domination number are equal, we show the Vizing-like inequality holds and γP(G□H)≥γP(G)γP(H).

On Transmission Irregular Cubic Graphs of an Arbitrary Order

The transmission of a vertex v of a graph G is the sum of distances from v to all the other vertices of G. A transmission irregular graph (TI graph) has mutually distinct vertex transmissions. In 2018, Alizadeh and Klavžar posed the following question: do there exist infinite families of regular TI graphs? An infinite family of TI cubic graphs of order 118+72k, k≥0, was constructed by Dobrynin in 2019. In this paper, we study the problem of finding TI cubic graphs for an arbitrary number of vertices. It is shown that there exists a TI cubic graph of an arbitrary even order n≥22. Almost all constructed graphs are contained in twelve infinite families.

Cubic graphs have paired-domination number at most four-seventh of their orders

A paired-dominating set of a graph G is a dominating set of vertices whose induced subgraph has a perfect matching. The paired-domination number γpr(G) is the minimum cardinality of a paired-dominating set of G. In this paper, we prove the conjecture of Goddard and Henning (2009) stating that if G is a connected cubic graph of order n and G is different from the Petersen graph, then γpr(G)≤4n/7.

Some results on the 3‐total‐rainbow index

In this article, we first discuss the 3-total rainbow index of unicyclic graphs. Moreover, we show a sharp upper bound for the 3-total-rainbow index of general graphs. Finally, we determine the 3-total-rainbow index of all small cubic graphs of order 8 or less.

Some results on the total properk-connection number

AbstractIn this paper, we first investigate the total proper connection number of a graphGGaccording to some constraints ofG¯\overline{G}. Next, we investigate the total proper connection numbers of graphGGwith large clique numberω(G)=n−s\omega \left(G)=n-sfor1≤s≤31\le s\le 3. Finally, we determine the total properkk-connection numbers of circular ladders, Möbius ladders and all small cubic graphs of order 8 or less.

Partially balanced incomplete block (PBIB)-designs associated with geodetic sets in graphs

A [Formula: see text]-design over regular graph [Formula: see text] of degree [Formula: see text] is an ordered pair [Formula: see text], where [Formula: see text] and [Formula: see text] is the set of geodetic sets in [Formula: see text] called blocks such that two vertices [Formula: see text] and [Formula: see text] which are [Formula: see text] associates occur together in [Formula: see text] blocks, the numbers [Formula: see text] being independent of the choice of the pair [Formula: see text] and [Formula: see text]. In this paper, we obtain Partially Balanced Incomplete Block (PBIB)-designs arising from geodetic sets in some classes of graphs. Also, we give a complete list of PBIB-designs with respect to the geodetic sets for cubic graphs of order at most [Formula: see text]. The discussion of non-existence of some designs corresponding to geodetic sets from certain graphs concludes the paper.

Domination versus total domination in claw-free cubic graphs

A set S of vertices in a graph G is a dominating set if every vertex not in S is adjacent to a vertex in S. If, in addition, every vertex in S is adjacent to some other vertex in S, then S is a total dominating set. The domination number γ(G) of G is the minimum cardinality of a dominating set in G, while the total domination number γt(G) of G is the minimum cardinality of total dominating set in G. A claw-free graph is a graph that does not contain K1,3 as an induced subgraph. Let G be a connected, claw-free, cubic graph of order n. We show that if we exclude two graphs, then γt(G)γ(G)≤127, and this bound is best possible. In order to prove this result, we prove that if we exclude four graphs, then γt(G)≤37n, and this bound is best possible. These bounds improve previously best known results due to Favaron and Henning (2008) [7], Southey and Henning (2010) [19].

The smallest pair of cospectral cubic graphs with different chromatic indexes

Using an exhaustive search on cubic graphs of order 16, we find a unique cospectral pair with different chromatic indexes. This example indicates that the chromatic index of a regular graph is not characterized by its spectrum, which answers a question recently posed in Etesami and Haemers (2020). We prove that any orthogonal matrix representing the similarity between the two adjacency matrices of the cospectral pair cannot be rational. This implies that the cospectral pair cannot be obtained using the original GM-switching method or its generalizations based on rational orthogonal matrices.

On (1,2)-domination in cubic graphs

A (1,2)-dominating set in a graph G with minimum degree at least 2 is a set S of vertices of G such that every vertex in V(G)∖S has at least one neighbor in S and every vertex in S has at least two neighbors in S. The (1,2)-domination number, γ1,2(G), of G is the minimum cardinality of a (1,2)-dominating set of G. In this paper we prove that if G is a cubic graph of order n, then γ1,2(G)≤34n, and this bound is tight.

Independent domination in subcubic graphs

A set S of vertices in a graph G is a dominating set if every vertex not in S is adjacent to a vertex in S. If, in addition, S is an independent set, then S is an independent dominating set. The independent domination number i(G) of G is the minimum cardinality of an independent dominating set in G. In Goddard and Henning (Discrete Math 313:839–854, 2013) conjectured that if G is a connected cubic graph of order n, then $$i(G) \le \frac{3}{8}n$$ , except if G is the complete bipartite graph $$K_{3,3}$$ or the 5-prism $$C_5 \, \Box \, K_2$$ . Further they construct two infinite families of connected cubic graphs with independent domination three-eighths their order. In this paper, we provide a new family of connected cubic graphs G of order n such that $$i(G) = \frac{3}{8}n$$ . We also show that if G is a subcubic graph of order n with no isolated vertex, then $$i(G) \le \frac{1}{2}n$$ , and we characterize the graphs achieving equality in this bound.

The spectrum problem for the connected cubic graphs of order 10

We show that if G is a connected cubic graph of order 10, then there exists a G-decomposition of Kv if and only if v≡1 or 10 (mod 15) except when v= 10 and G is one of 5 specific graphs.

Uniquely restricted matchings in subcubic graphs without short cycles

AbstractA matching in a graph is uniquely restricted if no other matching in covers the same set of vertices. We prove that any connected subcubic graph with vertices and girth at least 5 contains a uniquely restricted matching of size at least except for two exceptional cubic graphs of order 14 and 20.