Perfect graphs involving semitotal and semipaired domination

Let G be a graph with vertex set V and no isolated vertices, and let S be a dominating set of V. The set S is a semitotal dominating set of G if every vertex in S is within distance 2 of another vertex of S. And, S is a semipaired dominating set of G if S can be partitioned into 2-element subsets such that the vertices in each 2-set are at most distance two apart. The semitotal domination number \(\gamma _\mathrm{t2}(G)\) is the minimum cardinality of a semitotal dominating set of G, and the semipaired domination number \(\gamma _\mathrm{pr2}(G)\) is the minimum cardinality of a semipaired dominating set of G. For a graph without isolated vertices, the domination number \(\gamma (G)\), the total domination \(\gamma _t(G)\), and the paired domination number \(\gamma _\mathrm{pr}(G)\) are related to the semitotal and semipaired domination numbers by the following inequalities: \(\gamma (G) \le \gamma _\mathrm{t2}(G) \le \gamma _t(G) \le \gamma _\mathrm{pr}(G)\) and \(\gamma (G) \le \gamma _\mathrm{t2}(G) \le \gamma _\mathrm{pr2}(G) \le \gamma _\mathrm{pr}(G) \le 2\gamma (G)\). Given two graph parameters \(\mu \) and \(\psi \) related by a simple inequality \(\mu (G) \le \psi (G)\) for every graph G having no isolated vertices, a graph is \((\mu ,\psi )\)-perfect if every induced subgraph H with no isolated vertices satisfies \(\mu (H) = \psi (H)\). Alvarado et al. (Discrete Math 338:1424–1431, 2015) consider classes of \((\mu ,\psi )\)-perfect graphs, where \(\mu \) and \(\psi \) are domination parameters including \(\gamma \), \(\gamma _t\) and \(\gamma _\mathrm{pr}\). We study classes of perfect graphs for the possible combinations of parameters in the inequalities when \(\gamma _\mathrm{t2}\) and \(\gamma _\mathrm{pr2}\) are included in the mix. Our results are characterizations of several such classes in terms of their minimal forbidden induced subgraphs.     

The <Emphasis Type="Italic">k</Emphasis>-hop connected dominating set problem: approximation and hardness

Let G be a connected graph and k be a positive integer. A vertex subset D of G is a k-hop connected dominating set if the subgraph of G induced by D is connected, and for every vertex v in G there is a vertex u in D such that the distance between v and u in G is at most k. We study the problem of finding a minimum k-hop connected dominating set of a graph (\({\textsc {Min}}k{\hbox {-}\textsc {CDS}}\)). We prove that \({\textsc {Min}}k{\hbox {-}\textsc {CDS}}\) is \(\mathscr {NP}\)-hard on planar bipartite graphs of maximum degree 4. We also prove that \({\textsc {Min}}k{\hbox {-}\textsc {CDS}}\) is \(\mathscr {APX}\)-complete on bipartite graphs of maximum degree 4. We present inapproximability thresholds for \({\textsc {Min}}k{\hbox {-}\textsc {CDS}}\) on bipartite and on (1, 2)-split graphs. Interestingly, one of these thresholds is a parameter of the input graph which is not a function of its number of vertices. We also discuss the complexity of computing this graph parameter. On the positive side, we show an approximation algorithm for \({\textsc {Min}}k{\hbox {-}\textsc {CDS}}\). Finally, when \(k=1\), we present two new approximation algorithms for the weighted version of the problem restricted to graphs with a polynomially bounded number of minimal separators.     

Neighbor sum distinguishing list total coloring of subcubic graphs

Let \(G=(V, E)\) be a simple graph and denote the set of edges incident to a vertex v by E(v). The neighbor sum distinguishing (NSD) total choice number of G, denoted by \(\mathrm{ch}_{\Sigma }^{t}(G)\), is the smallest integer k such that, after assigning each \(z\in V\cup E\) a set L(z) of k real numbers, G has a total coloring \(\phi \) satisfying \(\phi (z)\in L(z)\) for each \(z\in V\cup E\) and \(\sum _{z\in E(u)\cup \{u\}}\phi (z)\ne \sum _{z\in E(v)\cup \{v\}}\phi (z)\) for each \(uv\in E\). In this paper, we propose some reducible configurations of NSD list total coloring for general graphs by applying the Combinatorial Nullstellensatz. As an application, we present that \(\mathrm{ch}^{t}_{\Sigma }(G)\le \Delta (G)+3\) for every subcubic graph G.     

Neighbor-sum-distinguishing edge choosability of subcubic graphs

A graph G is said to be neighbor-sum-distinguishing edge k-choose if, for every list L of colors such that L(e) is a set of k positive real numbers for every edge e, there exists a proper edge coloring which assigns to each edge a color from its list so that for each pair of adjacent vertices u and v the sum of colors taken on the edges incident to u is different from the sum of colors taken on the edges incident to v. Let \(\mathrm{ch}^{\prime }_{\sum ^p}(G)\) denote the smallest integer k such that G is neighbor-sum-distinguishing edge k-choose. In this paper, we prove that if G is a subcubic graph with the maximum average degree mad(G), then (1) \(\mathrm{ch}^{\prime }_{\sum ^p}(G)\le 7\); (2) \(\mathrm{ch}^{\prime }_{\sum ^p}(G)\le 6\) if \(\hbox {mad}(G)<\frac{36}{13}\); (3) \(\mathrm{ch}^{\prime }_{\sum ^p}(G)\le 5\) if \(\hbox {mad}(G)<\frac{5}{2}\).     

Online covering salesman problem

Given a graph \(G=(V,E,D,W)\), the generalized covering salesman problem (CSP) is to find a shortest tour in G such that each vertex \(i\in D\) is either on the tour or within a predetermined distance L to an arbitrary vertex \(j\in W\) on the tour, where \(D\subset V\),\(W\subset V\). In this paper, we propose the online CSP, where the salesman will encounter at most k blocked edges during the traversal. The edge blockages are real-time, meaning that the salesman knows about a blocked edge when it occurs. We present a lower bound \(\frac{1}{1 + (k + 2)L}k+1\) and a CoverTreeTraversal algorithm for online CSP which is proved to be \(k+\alpha \)-competitive, where \(\alpha =0.5+\frac{(4k+2)L}{OPT}+2\gamma \rho \), \(\gamma \) is the approximation ratio for Steiner tree problem and \(\rho \) is the maximal number of locations that a customer can be served. When \(\frac{L}{\texttt {OPT}}\rightarrow 0\), our algorithm is near optimal. The problem is also extended to the version with service cost, and similar results are derived.     

On the odd girth and the circular chromatic number of generalized Petersen graphs

A class \(\mathcal{G}\) of simple graphs is said to be girth-closed (odd-girth-closed) if for any positive integer g there exists a graph \(\mathrm {G} \in \mathcal{G}\) such that the girth (odd-girth) of \(\mathrm {G}\) is \(\ge g\). A girth-closed (odd-girth-closed) class \(\mathcal{G}\) of graphs is said to be pentagonal (odd-pentagonal) if there exists a positive integer \(g^*\) depending on \(\mathcal{G}\) such that any graph \(\mathrm {G} \in \mathcal{G}\) whose girth (odd-girth) is greater than \(g^*\) admits a homomorphism to the five cycle (i.e. is \(\mathrm {C}_{_{5}}\)-colourable). Although, the question “Is the class of simple 3-regular graphs pentagonal?” proposed by Ne?et?il (Taiwan J Math 3:381–423, 1999) is still a central open problem, Gebleh (Theorems and computations in circular colourings of graphs, 2007) has shown that there exists an odd-girth-closed subclass of simple 3-regular graphs which is not odd-pentagonal. In this article, motivated by the conjecture that the class of generalized Petersen graphs is odd-pentagonal, we show that finding the odd girth of generalized Petersen graphs can be transformed to an integer programming problem, and using the combinatorial and number theoretic properties of this problem, we explicitly compute the odd girth of such graphs, showing that the class is odd-girth-closed. Also, we obtain upper and lower bounds for the circular chromatic number of these graphs, and as a consequence, we show that the subclass containing generalized Petersen graphs \(\mathrm {Pet}(n,k)\) for which either k is even, n is odd and \(n\mathop {\equiv }\limits ^{k-1}\pm 2\) or both n and k are odd and \(n\ge 5k\) is odd-pentagonal. This in particular shows the existence of nontrivial odd-pentagonal subclasses of 3-regular simple graphs.     

Minimum 2-distance coloring of planar graphs and channel assignment

A 2-distance k-coloring of a graph G is a proper k-coloring such that any two vertices at distance two get different colors. \(\chi _{2}(G)\)=min{k|G has a 2-distance k-coloring}. Wegner conjectured that for each planar graph G with maximum degree \(\Delta \), \(\chi _2(G) \le 7\) if \(\Delta \le 3\), \(\chi _2(G) \le \Delta +5\) if \(4\le \Delta \le 7\) and \(\chi _2(G) \le \lfloor \frac{3\Delta }{2}\rfloor +1\) if \(\Delta \ge 8\). In this paper, we prove that: (1) If G is a planar graph with maximum degree \(\Delta \le 5\), then \(\chi _{2}(G)\le 20\); (2) If G is a planar graph with maximum degree \(\Delta \ge 6\), then \(\chi _{2}(G)\le 5\Delta -7\).     

Maximum coverage problem with group budget constraints

We study the maximum coverage problem with group budget constraints (MCG). The input consists of a ground set X, a collection \(\psi \) of subsets of X each of which is associated with a combinatorial structure such that for every set \(S_j\in \psi \), a cost \(c(S_j)\) can be calculated based on the combinatorial structure associated with \(S_j\), a partition \(G_1,G_2,\ldots ,G_l\) of \(\psi \), and budgets \(B_1,B_2,\ldots ,B_l\), and B. A solution to the problem consists of a subset H of \(\psi \) such that \(\sum _{S_j\in H} c(S_j) \le B\) and for each \(i \in {1,2,\ldots ,l}\), \(\sum _{S_j \in H\cap G_i}c(S_j)\le B_i\). The objective is to maximize \(|\bigcup _{S_j\in H}S_j|\). In our work we use a new and improved analysis of the greedy algorithm to prove that it is a \((\frac{\alpha }{3+2\alpha })\)-approximation algorithm, where \(\alpha \) is the approximation ratio of a given oracle which takes as an input a subset \(X^{new}\subseteq X\) and a group \(G_i\) and returns a set \(S_j\in G_i\) which approximates the optimal solution for \(\max _{D\in G_i}\frac{|D\cap X^{new}|}{c(D)}\). This analysis that is shown here to be tight for the greedy algorithm, improves by a factor larger than 2 the analysis of the best known approximation algorithm for MCG.     

Improved upper bound for the degenerate and star chromatic numbers of graphs

Let \(G=G(V,E)\) be a graph. A proper coloring of G is a function \(f:V\rightarrow N\) such that \(f(x)\ne f(y)\) for every edge \(xy\in E\). A proper coloring of a graph G such that for every \(k\ge 1\), the union of any k color classes induces a \((k-1)\)-degenerate subgraph is called a degenerate coloring; a proper coloring of a graph with no two-colored \(P_{4}\) is called a star coloring. If a coloring is both degenerate and star, then we call it a degenerate star coloring of graph. The corresponding chromatic number is denoted as \(\chi _{sd}(G)\). In this paper, we employ entropy compression method to obtain a new upper bound \(\chi _{sd}(G)\le \lceil \frac{19}{6}\Delta ^{\frac{3}{2}}+5\Delta \rceil \) for general graph G.     

Efficient reassembling of graphs,part 1: the linear case

The reassembling of a simple connected graph \(G = (V,E)\) is an abstraction of a problem arising in earlier studies of network analysis. Its simplest formulation is in two steps:

1. (1)
   We cut every edge of G into two halves, thus obtaining a collection of \(n = |\,V\,|\) one-vertex components, such that for every \(v\in V\) the one-vertex component \(\{ v \}\) has \({{degree}}_{}(v)\) half edges attached to it.
    
2. (2)
   We splice the two halves of every edge together, not of all the edges at once, but in some ordering \(\Theta \) of the edges that minimizes two measures that depend on the edge-boundary degrees of assembled components.
    

A component A is a subset of V and its edge-boundary degree is the number of edges in G with one endpoint in A and one endpoint in \(V-A\) (which is the same as the number of half edges attached to A after all edges with both endpoints in A have been spliced together). The maximum edge-boundary degree encountered during the reassembling process is what we call the \(\varvec{\alpha }\) -measure of the reassembling, and the sum of all edge-boundary degrees is its \(\varvec{\beta }\) -measure. The \(\alpha \)-optimization (resp. \(\beta \)-optimization) of the reassembling of G is to determine an order \(\Theta \) for splicing the edges that minimizes its \(\alpha \)-measure (resp. \(\beta \)-measure). There are different forms of reassembling, depending on restrictions and variations on the ordering \(\Theta \) of the edges. We consider only cases satisfying the condition that if an edge between disjoint components A and B is spliced, then all the edges between A and B are spliced at the same time. In this report, we examine the particular case of linear reassembling, which requires that the next edge to be spliced must be adjacent to an already spliced edge. We delay other forms of reassembling to follow-up reports. We prove that \(\alpha \)-optimization of linear reassembling and minimum-cutwidth linear arrangment (\(\mathrm{CutWidth}\)) are polynomially reducible to each other, and that \(\beta \)-optimization of linear reassembling and minimum-cost linear arrangement (\(\mathrm{MinArr}\)) are polynomially reducible to each other. The known NP-hardness of \(\mathrm{CutWidth}\) and \(\mathrm{MinArr}\) imply the NP-hardness of \(\alpha \)-optimization and \(\beta \)-optimization.

     

On minimally 2-connected graphs with generalized connectivity $$\kappa _{3}=2$$

For \(S\subseteq G\), let \(\kappa (S)\) denote the maximum number r of edge-disjoint trees \(T_1, T_2, \ldots , T_r\) in G such that \(V(T_i)\cap V(T_j)=S\) for any \(i,j\in \{1,2,\ldots ,r\}\) and \(i\ne j\). For every \(2\le k\le n\), the k-connectivity of G, denoted by \(\kappa _k(G)\), is defined as \(\kappa _k(G)=\hbox {min}\{\kappa (S)| S\subseteq V(G)\ and\ |S|=k\}\). Clearly, \(\kappa _2(G)\) corresponds to the traditional connectivity of G. In this paper, we focus on the structure of minimally 2-connected graphs with \(\kappa _{3}=2\). Denote by \(\mathcal {H}\) the set of minimally 2-connected graphs with \(\kappa _{3}=2\). Let \(\mathcal {B}\subseteq \mathcal {H}\) and every graph in \(\mathcal {B}\) is either \(K_{2,3}\) or the graph obtained by subdividing each edge of a triangle-free 3-connected graph. We obtain that \(H\in \mathcal {H}\) if and only if \(H\in \mathcal {B}\) or H can be constructed from one or some graphs \(H_{1},\ldots ,H_{k}\) in \(\mathcal {B}\) (\(k\ge 1\)) by applying some operations recursively.     

Upper bounds for adjacent vertex-distinguishing edge coloring

An adjacent vertex-distinguishing edge coloring of a graph is a proper edge coloring such that no pair of adjacent vertices meets the same set of colors. The adjacent vertex-distinguishing edge chromatic number is the minimum number of colors required for an adjacent vertex-distinguishing edge coloring, denoted as \(\chi '_{as}(G)\). In this paper, we prove that for a connected graph G with maximum degree \(\Delta \ge 3\), \(\chi '_{as}(G)\le 3\Delta -1\), which proves the previous upper bound. We also prove that for a graph G with maximum degree \(\Delta \ge 458\) and minimum degree \(\delta \ge 8\sqrt{\Delta ln \Delta }\), \(\chi '_{as}(G)\le \Delta +1+5\sqrt{\Delta ln \Delta }\).     

Nordhaus–Gaddum bounds for total Roman domination

A Nordhaus–Gaddum-type result is a lower or an upper bound on the sum or the product of a parameter of a graph and its complement. In this paper we continue the study of Nordhaus–Gaddum bounds for the total Roman domination number \(\gamma _{tR}\). Let G be a graph on n vertices and let \(\overline{G}\) denote the complement of G, and let \(\delta ^*(G)\) denote the minimum degree among all vertices in G and \(\overline{G}\). For \(\delta ^*(G)\ge 1\), we show that (i) if G and \(\overline{G}\) are connected, then \((\gamma _{tR}(G)-4)(\gamma _{tR}(\overline{G})-4)\le 4\delta ^*(G)-4\), (ii) if \(\gamma _{tR}(G), \gamma _{tR}(\overline{G})\ge 8\), then \(\gamma _{tR}(G)+\gamma _{tR}(\overline{G})\le 2\delta ^*(G)+5\) and (iii) \(\gamma _{tR}(G)+\gamma _{tR}(\overline{G})\le n+5\) and \(\gamma _{tR}(G)\gamma _{tR}(\overline{G})\le 6n-5\).     

A quadratic time exact algorithm for continuous connected 2-facility location problem in trees

This paper studies the continuous connected 2-facility location problem (CC2FLP) in trees. Let \(T = (V, E, c, d, \ell , \mu )\) be an undirected rooted tree, where each node \(v \in V\) has a weight \(d(v) \ge 0\) denoting the demand amount of v as well as a weight \(\ell (v) \ge 0\) denoting the cost of opening a facility at v, and each edge \(e \in E\) has a weight \(c(e) \ge 0\) denoting the cost on e and is associated with a function \(\mu (e,t) \ge 0\) denoting the cost of opening a facility at a point x(e, t) on e where t is a continuous variable on e. Given a subset \(\mathcal {D} \subseteq V\) of clients, and a subset \(\mathcal {F} \subseteq \mathcal {P}(T)\) of continuum points admitting facilities where \(\mathcal {P}(T)\) is the set of all the points on edges of T, when two facilities are installed at a pair of continuum points \(x_1\) and \(x_2\) in \(\mathcal {F}\), the total cost involved in CC2FLP includes three parts: the cost of opening two facilities at \(x_1\) and \(x_2\), K times the cost of connecting \(x_1\) and \(x_2\), and the cost of all the clients in \(\mathcal {D}\) connecting to some facility. The objective is to open two facilities at a pair of continuum points in \(\mathcal {F}\) to minimize the total cost, for a given input parameter \(K \ge 1\). This paper focuses on the case of \(\mathcal {D} = V\) and \(\mathcal {F} = \mathcal {P}(T)\). We first study the discrete version of CC2FLP, named the discrete connected 2-facility location problem (DC2FLP), where two facilities are restricted to the nodes of T, and devise a quadratic time edge-splitting algorithm for DC2FLP. Furthermore, we prove that CC2FLP is almost equivalent to DC2FLP in trees, and develop a quadratic time exact algorithm based on the edge-splitting algorithm. Finally, we adapt our algorithms to the general case of \(\mathcal {D} \subseteq V\) and \(\mathcal {F} \subseteq \mathcal {P}(T)\).     

On zero-sum $$\mathbb {Z}_{2j}^k$$-magic graphs

Let \(G = (V,E)\) be a finite graph and let \((\mathbb {A},+)\) be an abelian group with identity 0. Then G is \(\mathbb {A}\)-magic if and only if there exists a function \(\phi \) from E into \(\mathbb {A} - \{0\}\) such that for some \(c \in \mathbb {A}, \sum _{e \in E(v)} \phi (e) = c\) for every \(v \in V\), where E(v) is the set of edges incident to v. Additionally, G is zero-sum \(\mathbb {A}\)-magic if and only if \(\phi \) exists such that \(c = 0\). We consider zero-sum \(\mathbb {A}\)-magic labelings of graphs, with particular attention given to \(\mathbb {A} = \mathbb {Z}_{2j}^k\). For \(j \ge 1\), let \(\zeta _{2j}(G)\) be the smallest positive integer c such that G is zero-sum \(\mathbb {Z}_{2j}^c\)-magic if c exists; infinity otherwise. We establish upper bounds on \(\zeta _{2j}(G)\) when \(\zeta _{2j}(G)\) is finite, and show that \(\zeta _{2j}(G)\) is finite for all r-regular \(G, r \ge 2\). Appealing to classical results on the factors of cubic graphs, we prove that \(\zeta _4(G) \le 2\) for a cubic graph G, with equality if and only if G has no 1-factor. We discuss the problem of classifying cubic graphs according to the collection of finite abelian groups for which they are zero-sum group-magic.     

A sufficient condition for planar graphs with girth 5 to be (1, 7)-colorable

A graph G is \((d_1, d_2)\)-colorable if its vertices can be partitioned into subsets \(V_1\) and \(V_2\) such that in \(G[V_1]\) every vertex has degree at most \(d_1\) and in \(G[V_2]\) every vertex has degree at most \(d_2\). Let \(\mathcal {G}_5\) denote the family of planar graphs with minimum cycle length at least 5. It is known that every graph in \(\mathcal {G}_5\) is \((d_1, d_2)\)-colorable, where \((d_1, d_2)\in \{(2,6), (3,5),(4,4)\}\). We still do not know even if there is a finite positive d such that every graph in \(\mathcal {G}_5\) is (1, d)-colorable. In this paper, we prove that every graph in \(\mathcal {G}_5\) without adjacent 5-cycles is (1, 7)-colorable. This is a partial positive answer to a problem proposed by Choi and Raspaud that is every graph in \(\mathcal {G}_5\;(1, 7)\)-colorable?.     

Neighbor sum distinguishing total coloring of graphs with bounded treewidth

A proper total k-coloring \(\phi \) of a graph G is a mapping from \(V(G)\cup E(G)\) to \(\{1,2,\dots , k\}\) such that no adjacent or incident elements in \(V(G)\cup E(G)\) receive the same color. Let \(m_{\phi }(v)\) denote the sum of the colors on the edges incident with the vertex v and the color on v. A proper total k-coloring of G is called neighbor sum distinguishing if \(m_{\phi }(u)\not =m_{\phi }(v)\) for each edge \(uv\in E(G).\) Let \(\chi _{\Sigma }^t(G)\) be the neighbor sum distinguishing total chromatic number of a graph G. Pil?niak and Wo?niak conjectured that for any graph G, \(\chi _{\Sigma }^t(G)\le \Delta (G)+3\). In this paper, we show that if G is a graph with treewidth \(\ell \ge 3\) and \(\Delta (G)\ge 2\ell +3\), then \(\chi _{\Sigma }^t(G)\le \Delta (G)+\ell -1\). This upper bound confirms the conjecture for graphs with treewidth 3 and 4. Furthermore, when \(\ell =3\) and \(\Delta \ge 9\), we show that \(\Delta (G) + 1\le \chi _{\Sigma }^t(G)\le \Delta (G)+2\) and characterize graphs with equalities.     

Neighbor sum distinguishing total coloring of planar graphs without 4-cycles

Let \(G=(V,E)\) be a graph and \(\phi : V\cup E\rightarrow \{1,2,\ldots ,k\}\) be a proper total coloring of G. Let f(v) denote the sum of the color on a vertex v and the colors on all the edges incident with v. The coloring \(\phi \) is neighbor sum distinguishing if \(f(u)\ne f(v)\) for each edge \(uv\in E(G)\). The smallest integer k in such a coloring of G is the neighbor sum distinguishing total chromatic number of G, denoted by \(\chi _{\Sigma }''(G)\). Pil?niak and Wo?niak conjectured that \(\chi _{\Sigma }''(G)\le \Delta (G)+3\) for any simple graph. By using the famous Combinatorial Nullstellensatz, we prove that \(\chi _{\Sigma }''(G)\le \max \{\Delta (G)+2, 10\}\) for planar graph G without 4-cycles. The bound \(\Delta (G)+2\) is sharp if \(\Delta (G)\ge 8\).     

Base polyhedra and the linking property

An integer polyhedron \(P \subseteq {\mathbb {R}}^n\) has the linking property if for any \(f \in {\mathbb {Z}}^n\) and \(g \in {\mathbb {Z}}^n\) with \(f \le g\), P has an integer point between f and g if and only if it has both an integer point above f and an integer point below g. We prove that an integer polyhedron in the hyperplane \(\sum _{j=1}^n x_j=\beta \) is a base polyhedron if and only if it has the linking property. The result implies that an integer polyhedron has the strong linking property, as defined in Frank and Király (in: Cook, Lovász, Vygen (eds) Research trends in combinatorial optimization, Springer, Berlin, pp 87–126, 2009), if and only if it is a generalized polymatroid.     

List 2-distance $$\varDelta +3$$-coloring of planar graphs without 4,5-cycles

Let \(\chi _2(G)\) and \(\chi _2^l(G)\) be the 2-distance chromatic number and list 2-distance chromatic number of a graph G, respectively. Wegner conjectured that for each planar graph G with maximum degree \(\varDelta \) at least 4, \(\chi _2(G)\le \varDelta +5\) if \(4\le \varDelta \le 7\), and \(\chi _2(G)\le \lfloor \frac{3\varDelta }{2}\rfloor +1\) if \(\varDelta \ge 8\). Let G be a planar graph without 4,5-cycles. We show that if \(\varDelta \ge 26\), then \(\chi _2^l(G)\le \varDelta +3\). There exist planar graphs G with girth \(g(G)=6\) such that \(\chi _2^l(G)=\varDelta +2\) for arbitrarily large \(\varDelta \). In addition, we also discuss the list L(2, 1)-labeling number of G, and prove that \(\lambda _l(G)\le \varDelta +8\) for \(\varDelta \ge 27\).