Neighbor product distinguishing total colorings

A total-[k]-coloring of a graph G is a mapping \(\phi : V (G) \cup E(G)\rightarrow \{1, 2, \ldots , k\}\) such that any two adjacent elements in \(V (G) \cup E(G)\) receive different colors. Let f(v) denote the product of the color of a vertex v and the colors of all edges incident to v. A total-[k]-neighbor product distinguishing-coloring of G is a total-[k]-coloring of G such that \(f(u)\ne f(v)\), where \(uv\in E(G)\). By \(\chi ^{\prime \prime }_{\prod }(G)\), we denote the smallest value k in such a coloring of G. We conjecture that \(\chi _{\prod }^{\prime \prime }(G)\le \Delta (G)+3\) for any simple graph with maximum degree \(\Delta (G)\). In this paper, we prove that the conjecture holds for complete graphs, cycles, trees, bipartite graphs and subcubic graphs. Furthermore, we show that if G is a \(K_4\)-minor free graph with \(\Delta (G)\ge 4\), then \(\chi _{\prod }^{\prime \prime }(G)\le \Delta (G)+2\).     

Neighbor sum distinguishing total coloring of 2-degenerate graphs

A proper k-total coloring of a graph G is a mapping from \(V(G)\cup E(G)\) to \(\{1,2,\ldots ,k\}\) such that no two adjacent or incident elements in \(V(G)\cup E(G)\) receive the same color. Let f(v) denote the sum of the colors on the edges incident with v and the color on vertex v. A proper k-total coloring of G is called neighbor sum distinguishing if \(f(u)\ne f(v)\) for each edge \(uv\in E(G)\). Let \(\chi ''_{\Sigma }(G)\) denote the smallest integer k in such a coloring of G. Pil?niak and Wo?niak conjectured that for any graph G, \(\chi ''_{\Sigma }(G)\le \Delta (G)+3\). In this paper, we show that if G is a 2-degenerate graph, then \(\chi ''_{\Sigma }(G)\le \Delta (G)+3\); Moreover, if \(\Delta (G)\ge 5\) then \(\chi ''_{\Sigma }(G)\le \Delta (G)+2\).     

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.     

Total and forcing total edge-to-vertex monophonic number of a graph

For a connected graph \(G = \left( V,E\right) \), a set \(S\subseteq E(G)\) is called a total edge-to-vertex monophonic set of a connected graph G if the subgraph induced by S has no isolated edges. The total edge-to-vertex monophonic number \(m_{tev}(G)\) of G is the minimum cardinality of its total edge-to-vertex monophonic set of G. The total edge-to-vertex monophonic number of certain classes of graphs is determined and some of its general properties are studied. Connected graphs of size \(q \ge 3 \) with total edge-to-vertex monophonic number q is characterized. It is shown that for positive integers \(r_{m},d_{m}\) and \(l\ge 4\) with \(r_{m}< d_{m} \le 2 r_{m}\), there exists a connected graph G with \(\textit{rad}_ {m} G = r_{m}\), \(\textit{diam}_ {m} G = d_{m}\) and \(m_{tev}(G) = l\) and also shown that for every integers a and b with \(2 \le a \le b\), there exists a connected graph G such that \( m_{ev}\left( G\right) = b\) and \(m_{tev}(G) = a + b\). A forcing subset for S of minimum cardinality is a minimum forcing subset of S. The forcing total edge-to-vertex monophonic number of S, denoted by \(f_{tev}(S)\) is the cardinality of a minimum forcing subset of S. The forcing total edge-to-vertex monophonic number of G, denoted by \(f_{tev}(G) = \textit{min}\{f_{tev}(S)\}\), where the minimum is taken over all total edge-to-vertex monophonic set S in G. The forcing total edge-to-vertex monophonic number of certain classes of graphs are determined and some of its general properties are studied. It is shown that for every integers a and b with \(0 \le a \le b\) and \(b \ge 2\), there exists a connected graph G such that \(f_{tev}(G) = a\) and \( m _{tev}(G) = b\), where \( f _{tev}(G)\) is the forcing total edge-to-vertex monophonic number of G.     

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.     

The $$(k,\ell )$$-proper index of graphs

A tree T in an edge-colored graph is called a proper tree if no two adjacent edges of T receive the same color. Let G be a connected graph of order n and k be an integer with \(2\le k \le n\). For \(S\subseteq V(G)\) and \(|S| \ge 2\), an S-tree is a tree containing the vertices of S in G. A set \(\{T_1,T_2,\ldots ,T_\ell \}\) of S-trees is called internally disjoint if \(E(T_i)\cap E(T_j)=\emptyset \) and \(V(T_i)\cap V(T_j)=S\) for \(1\le i\ne j\le \ell \). For a set S of k vertices of G, the maximum number of internally disjoint S-trees in G is denoted by \(\kappa (S)\). The k-connectivity \(\kappa _k(G)\) of G is defined by \(\kappa _k(G)=\min \{\kappa (S)\mid S\) is a k-subset of \(V(G)\}\). For a connected graph G of order n and for two integers k and \(\ell \) with \(2\le k\le n\) and \(1\le \ell \le \kappa _k(G)\), the \((k,\ell )\)-proper index \(px_{k,\ell }(G)\) of G is the minimum number of colors that are required in an edge-coloring of G such that for every k-subset S of V(G), there exist \(\ell \) internally disjoint proper S-trees connecting them. In this paper, we show that for every pair of positive integers k and \(\ell \) with \(k \ge 3\) and \(\ell \le \kappa _k(K_{n,n})\), there exists a positive integer \(N_1=N_1(k,\ell )\) such that \(px_{k,\ell }(K_n) = 2\) for every integer \(n \ge N_1\), and there exists also a positive integer \(N_2=N_2(k,\ell )\) such that \(px_{k,\ell }(K_{m,n}) = 2\) for every integer \(n \ge N_2\) and \(m=O(n^r) (r \ge 1)\). In addition, we show that for every \(p \ge c\root k \of {\frac{\log _a n}{n}}\) (\(c \ge 5\)), \(px_{k,\ell }(G_{n,p})\le 2\) holds almost surely, where \(G_{n,p}\) is the Erd?s–Rényi random graph model.     

Majorization and the spectral radius of starlike trees

A starlike tree is a tree with exactly one vertex of degree greater than two. The spectral radius of a graph G, that is denoted by \(\lambda (G)\), is the largest eigenvalue of G. Let k and \(n_1,\ldots ,n_k\) be some positive integers. Let \(T(n_1,\ldots ,n_k)\) be the tree T (T is a path or a starlike tree) such that T has a vertex v so that \(T{\setminus } v\) is the disjoint union of the paths \(P_{n_1-1},\ldots ,P_{n_k-1}\) where every neighbor of v in T has degree one or two. Let \(P=(p_1,\ldots ,p_k)\) and \(Q=(q_1,\ldots ,q_k)\), where \(p_1\ge \cdots \ge p_k\ge 1\) and \(q_1\ge \cdots \ge q_k\ge 1\) are integer. We say P majorizes Q and let \(P\succeq _M Q\), if for every j, \(1\le j\le k\), \(\sum _{i=1}^{j}p_i\ge \sum _{i=1}^{j}q_i\), with equality if \(j=k\). In this paper we show that if P majorizes Q, that is \((p_1,\ldots ,p_k)\succeq _M(q_1,\ldots ,q_k)\), then \(\lambda (T(q_1,\ldots ,q_k))\ge \lambda (T(p_1,\ldots ,p_k))\).     

Sparse multipartite graphs as partition universal for graphs with bounded degree

For graphs G and H, let \(G\rightarrow (H,H)\) signify that any red/blue edge coloring of G contains a monochromatic H as a subgraph. Denote \(\mathcal {H}(\Delta ,n)=\{H:|V(H)|=n,\Delta (H)\le \Delta \}\). For any \(\Delta \) and n, we say that G is partition universal for \(\mathcal {H}(\Delta ,n)\) if \(G\rightarrow (H,H)\) for every \(H\in \mathcal {H}(\Delta ,n)\). Let \(G_r(N,p)\) be the random spanning subgraph of the complete r-partite graph \(K_r(N)\) with N vertices in each part, in which each edge of \(K_r(N)\) appears with probability p independently and randomly. We prove that for fixed \(\Delta \ge 2\) there exist constants r, B and C depending only on \(\Delta \) such that if \(N\ge Bn\) and \(p=C(\log N/N)^{1/\Delta }\), then asymptotically almost surely \(G_r(N,p)\) is partition universal for \(\mathcal {H}(\Delta ,n)\).     

Paired-domination number of claw-free odd-regular graphs

A paired-dominating set of a graph G is a dominating set of vertices whose induced subgraph has a perfect matching, while the paired-domination number is the minimum cardinality of a paired-dominating set in the graph, denoted by \(\gamma _{pr}(G)\). Let G be a connected \(\{K_{1,3}, K_{4}-e\}\)-free cubic graph of order n. We show that \(\gamma _{pr}(G)\le \frac{10n+6}{27}\) if G is \(C_{4}\)-free and that \(\gamma _{pr}(G)\le \frac{n}{3}+\frac{n+6}{9(\lceil \frac{3}{4}(g_o+1)\rceil +1)}\) if G is \(\{C_{4}, C_{6}, C_{10}, \ldots , C_{2g_o}\}\)-free for an odd integer \(g_o\ge 3\); the extremal graphs are characterized; we also show that if G is a 2 -connected, \(\gamma _{pr}(G) = \frac{n}{3} \). Furthermore, if G is a connected \((2k+1)\)-regular \(\{K_{1,3}, K_4-e\}\)-free graph of order n, then \(\gamma _{pr}(G)\le \frac{n}{k+1} \), with equality if and only if \(G=L(F)\), where \(F\cong K_{1, 2k+2}\), or k is even and \(F\cong K_{k+1,k+2}\).     

Planar graphs without chordal 5-cycles are 2-good

Let G be a connected graph with \(n\ge 2\) vertices. Suppose that a fire breaks out at a vertex v of G. A firefighter starts to protect vertices. At each step, the firefighter protects two vertices not yet on fire. At the end of each step, the fire spreads to all the unprotected vertices that have a neighbour on fire. Let sn\(_2(v)\) denote the maximum number of vertices in G that the firefighter can save when a fire breaks out at vertex v. The 2-surviving rate \(\rho _2(G)\) of G is defined to be the real number \(\frac{1}{n^2} \sum _{v\in V(G)} \mathrm{sn}_2(v)\). Then it is obvious that \(0<\rho _2(G)<1\). The graph G is called 2-good if there is a constant \(c>0\) such that \(\rho _2(G)>c\). In this paper, we prove that every planar graph with \(n\ge 2\) vertices and without chordal 5-cycles is 2-good.     

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.     

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.     

The game chromatic index of some trees with maximum degree four and adjacent degree-four vertices

A (proper) total-k-coloring of a graph G is a mapping \(\phi : V (G) \cup E(G)\mapsto \{1, 2, \ldots , k\}\) such that any two adjacent or incident elements in \(V (G) \cup E(G)\) receive different colors. Let C(v) denote the set of the color of a vertex v and the colors of all incident edges of v. An adjacent vertex distinguishing total-k-coloring of G is a total-k-coloring of G such that for each edge \(uv\in E(G)\), \(C(u)\ne C(v)\). We denote the smallest value k in such a coloring of G by \(\chi ^{\prime \prime }_{a}(G)\). It is known that \(\chi _{a}^{\prime \prime }(G)\le \Delta (G)+3\) for any planar graph with \(\Delta (G)\ge 10\). In this paper, we consider the list version of this coloring and show that if G is a planar graph with \(\Delta (G)\ge 11\), then \({ ch}_{a}^{\prime \prime }(G)\le \Delta (G)+3\), where \({ ch}^{\prime \prime }_a(G)\) is the adjacent vertex distinguishing total choosability.     

On the difference of two generalized connectivities of a graph

The concept of k-connectivity \(\kappa '_{k}(G)\) of a graph G, introduced by Chartrand in 1984, is a generalization of the cut-version of the classical connectivity. Another generalized connectivity of a graph G, named the generalized k-connectivity \(\kappa _{k}(G)\), mentioned by Hager in 1985, is a natural generalization of the path-version of the classical connectivity. In this paper, we get the lower and upper bounds for the difference of these two parameters by showing that for a connected graph G of order n, if \(\kappa '_k(G)\ne n-k+1\) where \(k\ge 3\), then \(0\le \kappa '_k(G)-\kappa _k(G)\le n-k-1\); otherwise, \(-\lfloor \frac{k}{2}\rfloor +1\le \kappa '_k(G)-\kappa _k(G)\le n-k\). Moreover, all of these bounds are sharp. Some specific study is focused for the case \(k=3\). As results, we characterize the graphs with \(\kappa '_3(G)=\kappa _3(G)=t\) for \(t\in \{1, n-3, n-2\}\), and give a necessary condition for \(\kappa '_3(G)=\kappa _3(G)\) by showing that for a connected graph G of order n and size m, if \(\kappa '_3(G)=\kappa _3(G)=t\) where \(1\le t\le n-3\), then \(m\le {n-2\atopwithdelims ()2}+2t\). Moreover, the unique extremal graph is given for the equality to hold.     

Algorithmic aspects of <Emphasis Type="Italic">b</Emphasis>-disjunctive domination in graphs

For a fixed integer \(b>1\), a set \(D\subseteq V\) is called a b-disjunctive dominating set of the graph \(G=(V,E)\) if for every vertex \(v\in V{\setminus }D\), v is either adjacent to a vertex of D or has at least b vertices in D at distance 2 from it. The Minimum b-Disjunctive Domination Problem (MbDDP) is to find a b-disjunctive dominating set of minimum cardinality. The cardinality of a minimum b-disjunctive dominating set of G is called the b-disjunctive domination number of G, and is denoted by \(\gamma _{b}^{d}(G)\). Given a positive integer k and a graph G, the b-Disjunctive Domination Decision Problem (bDDDP) is to decide whether G has a b-disjunctive dominating set of cardinality at most k. In this paper, we first show that for a proper interval graph G, \(\gamma _{b}^{d}(G)\) is equal to \(\gamma (G)\), the domination number of G for \(b \ge 3\) and observe that \(\gamma _{b}^{d}(G)\) need not be equal to \(\gamma (G)\) for \(b=2\). We then propose a polynomial time algorithm to compute a minimum cardinality b-disjunctive dominating set of a proper interval graph for \(b=2\). Next we tighten the NP-completeness of bDDDP by showing that it remains NP-complete even in chordal graphs. We also propose a \((\ln ({\varDelta }^{2}+(b-1){\varDelta }+b)+1)\)-approximation algorithm for MbDDP, where \({\varDelta }\) is the maximum degree of input graph \(G=(V,E)\) and prove that MbDDP cannot be approximated within \((1-\epsilon ) \ln (|V|)\) for any \(\epsilon >0\) unless NP \(\subseteq \) DTIME\((|V|^{O(\log \log |V|)})\). Finally, we show that MbDDP is APX-complete for bipartite graphs with maximum degree \(\max \{b,4\}\).     

First-Fit colorings of graphs with no cycles of a prescribed even length

The First-Fit (or Grundy) chromatic number of a graph G denoted by \(\chi _{{_\mathsf{FF}}}(G)\), is the maximum number of colors used by the First-Fit (greedy) coloring algorithm when applied to G. In this paper we first show that any graph G contains a bipartite subgraph of Grundy number \(\lfloor \chi _{{_\mathsf{FF}}}(G) /2 \rfloor +1\). Using this result we prove that for every \(t\ge 2\) there exists a real number \(c>0\) such that in every graph G on n vertices and without cycles of length 2t, any First-Fit coloring of G uses at most \(cn^{1/t}\) colors. It is noted that for \(t=2\) this bound is the best possible. A compactness conjecture is also proposed concerning the First-Fit chromatic number involving the even girth of graphs.     

On the algorithmic aspects of strong subcoloring

A partition of the vertex set V(G) of a graph G into \(V(G)=V_1\cup V_2\cup \cdots \cup V_k\) is called a k-strong subcoloring if \(d(x,y)\ne 2\) in G for every \(x,y\in V_i\), \(1\le i \le k\) where d(x, y) denotes the length of a shortest x-y path in G. The strong subchromatic number is defined as \(\chi _{sc}(G)=\text {min}\{ k:G \text { admits a }k\)-\(\text {strong subcoloring}\}\). In this paper, we explore the complexity status of the StrongSubcoloring problem: for a given graph G and a positive integer k, StrongSubcoloring is to decide whether G admits a k-strong subcoloring. We prove that StrongSubcoloring is NP-complete for subcubic bipartite graphs and the problem is polynomial time solvable for trees. In addition, we prove the following dichotomy results: (i) for the class of \(K_{1,r}\)-free split graphs, StrongSubcoloring is in P when \(r\le 3\) and NP-complete when \(r>3\) and (ii) for the class of H-free graphs, StrongSubcoloring is polynomial time solvable only if H is an induced subgraph of \(P_4\); otherwise the problem is NP-complete. Next, we consider a lower bound on the strong subchromatic number. A strong set is a set S of vertices of a graph G such that for every \(x,y\in S\), \(d(x,y)= 2\) in G and the cardinality of a maximum strong set in G is denoted by \(\alpha _{s}(G)\). Clearly, \(\alpha _{s}(G)\le \chi _{sc}(G)\). We consider the complexity status of the StrongSet problem: given a graph G and a positive integer k, StrongSet asks whether G contains a strong set of cardinality k. We prove that StrongSet is NP-complete for (i) bipartite graphs and (ii) \(K_{1,4}\)-free split graphs, and it is polynomial time solvable for (i) trees and (ii) \(P_4\)-free graphs.     

Neighbor sum distinguishing total choosability of planar graphs

A total-k-coloring of a graph G is a mapping \(c: V(G)\cup E(G)\rightarrow \{1, 2,\dots , k\}\) such that any two adjacent or incident elements in \(V(G)\cup E(G)\) receive different colors. For a total-k-coloring of G, let \(\sum _c(v)\) denote the total sum of colors of the edges incident with v and the color of v. If for each edge \(uv\in E(G)\), \(\sum _c(u)\ne \sum _c(v)\), then we call such a total-k-coloring neighbor sum distinguishing. The least number k needed for such a coloring of G is the neighbor sum distinguishing total chromatic number, denoted by \(\chi _{\Sigma }^{''}(G)\). Pil?niak and Wo?niak conjectured \(\chi _{\Sigma }^{''}(G)\le \Delta (G)+3\) for any simple graph with maximum degree \(\Delta (G)\). In this paper, we prove that for any planar graph G with maximum degree \(\Delta (G)\), \(ch^{''}_{\Sigma }(G)\le \max \{\Delta (G)+3,16\}\), where \(ch^{''}_{\Sigma }(G)\) is the neighbor sum distinguishing total choosability of G.     

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.     

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 \(\varDelta \ge 27\).