## Abstract

A graph *G* is called *r-spanning cyclable* if for every *r* distinct vertices *v*
_{1}, *v*
_{2}, . *. . , v*
* _{r}* of

*G*, there exists

*r*cycles

*C*

_{1},

*C*

_{2}, .

*. . , C*

*in*

_{r}*G*such that

*v*

*is on*

_{i}*C*

*for every*

_{i}*i*, and every vertex of

*G*is on exactly one cycle

*C*

*. In this paper, we consider the 2-spanning cyclable problem for the generalized Petersen graph*

_{i}*GP*(

*n, k*). We solved the problem for

*k ≤*4. In addition, we provide an additional observation for general

*k*as well as stating a conjecture.

## Abstract

An *additive coloring* of a graph *G* is a labeling of the vertices of *G* from {1, 2, . *. . , k*} such that two adjacent vertices have distinct sums of labels on their neighbors. The least integer *k* for which a graph *G* has an additive coloring is called the *additive coloring number* of *G*, denoted *χ*
_{Σ} (*G*). Additive coloring is also studied under the names lucky labeling and open distinguishing. In this paper, we improve the current bounds on the additive coloring number for particular classes of graphs by proving results for a list version of additive coloring. We apply the discharging method and the Combinatorial Nullstellensatz to show that every planar graph *G* with girth at least 5 has *χ*
_{Σ} (*G*) ≤ 19, and for girth at least 6, 7, and 26, *χ*
_{Σ} (*G*) is at most 9, 8, and 3, respectively. In 2009, Czerwiński, Grytczuk, and Żelazny conjectured that *χ*
_{Σ} (*G*) ≤ *χ*(*G*), where *χ*(*G*) is the chromatic number of *G*. Our result for the class of non-bipartite planar graphs of girth at least 26 is best possible and affirms the conjecture for this class of graphs.

## Abstract

Let *H* be a graph. A decomposition of *H* is a set of edge-disjoint subgraphs of *H* whose union is *H*. A Hamiltonian path (respectively, cycle) of *H* is a path (respectively, cycle) that contains every vertex of *H* exactly once. A *k*-star, denoted by *S*
* _{k}*, is a star with

*k*edges. In this paper, we give necessary and sufficient conditions for decomposing the complete graph into α copies of Hamiltonian path (cycle) and

*β*copies of

*S*

_{3}.

## Abstract

The edit distance function of a hereditary property is the asymptotically largest edit distance between a graph of density *p* ∈ [0, 1] and . Denote by *P*
* _{n}* and

*C*

*the path graph of order*

_{n}*n*and the cycle graph of order

*n*, respectively. Let

*C*

_{2}

*with a diagonal, and*

_{n}*v*

_{0},

*v*

_{1}, .

*. . , v*

_{n}_{−1}} and

*C*

*, while Berikkyzy*

_{n}*et al.*studied the edit distance of powers of cycles. In this paper, by using the methods of Peck and Martin, we determine the edit distance function of

*P*

*, respectively.*

_{n}## Abstract

For an integer *k* at least 2, and a graph *G*, let *f*
* _{k}*(

*G*) be the minimum cardinality of a set

*X*of vertices of

*G*such that

*G*−

*X*has either

*k*vertices of maximum degree or order less than

*k*. Caro and Yuster [Discrete Mathematics 310 (2010) 742–747] conjectured that, for every

*k*, there is a constant

*c*

*such that*

_{k}*G*. Verifying a conjecture of Caro, Lauri, and Zarb [arXiv:1704.08472v1], we show the best possible result that, if

*t*is a positive integer, and

*F*is a forest of order at most

*f*

_{2}(

*F*) ≤

*t*. We study

*f*

_{3}(

*F*) for forests

*F*in more detail obtaining similar almost tight results, and we establish upper bounds on

*f*

*(*

_{k}*G*) for graphs

*G*of girth at least 5. For graphs

*G*of girth more than 2

*p*, for

*p*at least 3, our results imply

*k*, and every given forest

*F*, the value of

*f*

*(*

_{k}*F*) can be determined in polynomial time.

## Abstract

A Roman dominating function on a graph *G* = (*V, E*) is a function *f*:*V* (*G*) → {0, 1, 2} such that every vertex *u* for which *f*(*u*) = 0 is adjacent to at least one vertex *v* with *f*(*v*) = 2. The weight of a Roman dominating function is the value *w*(*f*) = Σ_{u}_{∈}
_{V(G)}*f*(*u*). The minimum weight of a Roman dominating function on a graph *G* is called the Roman domination number of *G*, denoted by γ* _{R}*(

*G*). In 2009, Chambers, Kinnersley, Prince and West proved that for any graph

*G*with

*n*vertices and maximum degree Δ, γ

*(*

_{R}*G*) ≤

*n*+ 1 − Δ. In this paper, we give a characterization of graphs attaining the previous bound including trees, regular and semiregular graphs. Moreover, we prove that the problem of deciding whether γ

*(*

_{R}*G*) =

*n*+ 1 − Δ is co-𝒩 𝒫-complete. Finally, we provide a characterization of extremal graphs of a Nordhaus–Gaddum bound for γ

*(*

_{R}*G*) + γ

*(*

_{R}*Ḡ*), where

*Ḡ*is the complement graph of

*G.*