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.     

L(2,1)-labelings of the edge-multiplicity-paths-replacement of a graph

An L(2,1)-labeling of a graph \(G\) is an assignment of nonnegative integers to \(V(G)\) such that the difference between labels of adjacent vertices is at least \(2\), and the difference between labels of vertices that are distance two apart is at least 1. The span of an L(2,1)-labeling of a graph \(G\) is the difference between the maximum and minimum integers used by it. The minimum span of an L(2,1)-labeling of \(G\) is denoted by \(\lambda (G)\). This paper focuses on L(2,1)-labelings-number of the edge-multiplicity-paths-replacement \(G(rP_{k})\) of a graph \(G\). In this paper, we obtain that \( r\Delta +1 \le \lambda (G(rP_{5}))\le r\Delta +2\), \(\lambda (G(rP_{k}))= r\Delta +1\) for \(k\ge 6\); and \(\lambda (G(rP_{4}))\le (\Delta +1)r+1\), \(\lambda (G(rP_{3}))\le (\Delta +1)r+\Delta \) for any graph \(G\) with maximum degree \(\Delta \). And the L(2,1)-labelings-numbers of the edge-multiplicity-paths-replacement \(G(rP_{k})\) are completely determined for \(1\le \Delta \le 2\). And we show that the class of graphs \(G(rP_{k})\) with \(k\ge 3 \) satisfies the conjecture: \(\lambda ^{T}_{2}(G)\le \Delta +2\) by Havet and Yu (Technical Report 4650, 2002).     

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.     

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.     

Pac king spanning trees and spanning 2-connected k-edge-connected essentially $$(2 k-1)$$(2 k-1)-edge-connected subgraphs

Let \(k\ge 2, p\ge 1, q\ge 0\) be integers. We prove that every \((4kp-2p+2q)\)-connected graph contains p spanning subgraphs \(G_i\) for \(1\le i\le p\) and q spanning trees such that all \(p+q\) subgraphs are pairwise edge-disjoint and such that each \(G_i\) is k-edge-connected, essentially \((2k-1)\)-edge-connected, and \(G_i -v\) is \((k-1)\)-edge-connected for all \(v\in V(G)\). This extends the well-known result of Nash-Williams and Tutte on packing spanning trees, a theorem that every 6p-connected graph contains p pairwise edge-disjoint spanning 2-connected subgraphs, and a theorem that every \((6p+2q)\)-connected graph contains p spanning 2-connected subgraphs and q spanning trees, which are all pairwise edge-disjoint. As an application, we improve a result on k-arc-connected orientations.     

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.     

On the complete width and edge clique cover problems

A complete graph is the graph in which every two vertices are adjacent. For a graph \(G=(V,E)\), the complete width of G is the minimum k such that there exist k independent sets \(\mathtt {N}_i\subseteq V\), \(1\le i\le k\), such that the graph \(G'\) obtained from G by adding some new edges between certain vertices inside the sets \(\mathtt {N}_i\), \(1\le i\le k\), is a complete graph. The complete width problem is to decide whether the complete width of a given graph is at most k or not. In this paper we study the complete width problem. We show that the complete width problem is NP-complete on \(3K_2\)-free bipartite graphs and polynomially solvable on \(2K_2\)-free bipartite graphs and on \((2K_2,C_4)\)-free graphs. As a by-product, we obtain the following new results: the edge clique cover problem is NP-complete on \(\overline{3K_2}\)-free co-bipartite graphs and polynomially solvable on \(C_4\)-free co-bipartite graphs and on \((2K_2, C_4)\)-free graphs. We also give a characterization for k-probe complete graphs which implies that the complete width problem admits a kernel of at most \(2^k\) vertices. This provides another proof for the known fact that the edge clique cover problem admits a kernel of at most \(2^k\) vertices. Finally we determine all graphs of small complete width \(k\le 3\).     

2-Edge connected dominating sets and 2-Connected dominating sets of a graph

A \(k\)-connected (resp. \(k\)-edge connected) dominating set \(D\) of a connected graph \(G\) is a subset of \(V(G)\) such that \(G[D]\) is \(k\)-connected (resp. \(k\)-edge connected) and each \(v\in V(G)\backslash D\) has at least one neighbor in \(D\). The \(k\) -connected domination number (resp. \(k\) -edge connected domination number) of a graph \(G\) is the minimum size of a \(k\)-connected (resp. \(k\)-edge connected) dominating set of \(G\), and denoted by \(\gamma _k(G)\) (resp. \(\gamma '_k(G)\)). In this paper, we investigate the relation of independence number and 2-connected (resp. 2-edge-connected) domination number, and prove that for a graph \(G\), if it is \(2\)-edge connected, then \(\gamma '_2(G)\le 4\alpha (G)-1\), and it is \(2\)-connected, then \(\gamma _2(G)\le 6\alpha (G)-3\), where \(\alpha (G)\) is the independent number of \(G\).     

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 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.     

Planar graphs without 4-cycles and close triangles are (2, 0, 0)-colorable

For a set of nonnegative integers \(c_1, \ldots , c_k\), a \((c_1, c_2,\ldots , c_k)\)-coloring of a graph G is a partition of V(G) into \(V_1, \ldots , V_k\) such that for every i, \(1\le i\le k, G[V_i]\) has maximum degree at most \(c_i\). We prove that all planar graphs without 4-cycles and no less than two edges between triangles are (2, 0, 0)-colorable.     

The (vertex-)monochromatic index of a graph

A tree T in an edge-colored (vertex-colored) graph H is called a monochromatic (vertex-monochromatic) tree if all the edges (internal vertices) of T have the same color. For \(S\subseteq V(H)\), a monochromatic (vertex-monochromatic) S-tree in H is a monochromatic (vertex-monochromatic) tree of H containing the vertices of S. For a connected graph G and a given integer k with \(2\le k\le |V(G)|\), the k -monochromatic index \(mx_k(G)\) (k -monochromatic vertex-index \(mvx_k(G)\)) of G is the maximum number of colors needed such that for each subset \(S\subseteq V(G)\) of k vertices, there exists a monochromatic (vertex-monochromatic) S-tree. For \(k=2\), Caro and Yuster showed that \(mc(G)=mx_2(G)=|E(G)|-|V(G)|+2\) for many graphs, but it is not true in general. In this paper, we show that for \(k\ge 3\), \(mx_k(G)=|E(G)|-|V(G)|+2\) holds for any connected graph G, completely determining the value. However, for the vertex-version \(mvx_k(G)\) things will change tremendously. We show that for a given connected graph G, and a positive integer L with \(L\le |V(G)|\), to decide whether \(mvx_k(G)\ge L\) is NP-complete for each integer k such that \(2\le k\le |V(G)|\). Finally, we obtain some Nordhaus–Gaddum-type results for the k-monochromatic vertex-index.     

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 \(F\cong K_{k+1,k+2}\).     

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.     

Distance domination in graphs with given minimum and maximum degree

For an integer \(k \ge 1\), a distance k-dominating set of a connected graph G is a set S of vertices of G such that every vertex of V(G) is at distance at most k from some vertex of S. The distance k-domination number \(\gamma _k(G)\) of G is the minimum cardinality of a distance k-dominating set of G. In this paper, we establish an upper bound on the distance k-domination number of a graph in terms of its order, minimum degree and maximum degree. We prove that for \(k \ge 2\), if G is a connected graph with minimum degree \(\delta \ge 2\) and maximum degree \(\Delta \) and of order \(n \ge \Delta + k - 1\), then \(\gamma _k(G) \le \frac{n + \delta - \Delta }{\delta + k - 1}\). This result improves existing known results.     

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 \(\varDelta \ge 27\).     

Embeddings into almost self-centered graphs of given radius

A graph is almost self-centered (ASC) if all but two of its vertices are central. An almost self-centered graph with radius r is called an r-ASC graph. The r-ASC index \(\theta _r(G)\) of a graph G is the minimum number of vertices needed to be added to G such that an r-ASC graph is obtained that contains G as an induced subgraph. It is proved that \(\theta _r(G)\le 2r\) holds for any graph G and any \(r\ge 2\) which improves the earlier known bound \(\theta _r(G)\le 2r+1\). It is further proved that \(\theta _r(G)\le 2r-1\) holds if \(r\ge 3\) and G is of order at least 2. The 3-ASC index of complete graphs is determined. It is proved that \(\theta _3(G)\in \{3,4\}\) if G has diameter 2 and for several classes of graphs of diameter 2 the exact value of the 3-ASC index is obtained. For instance, if a graph G of diameter 2 does not contain a diametrical triple, then \(\theta _3(G) = 4\). The 3-ASC index of paths of order \(n\ge 1\), cycles of order \(n\ge 3\), and trees of order \(n\ge 10\) and diameter \(n-2\) are also determined, respectively, and several open problems proposed.     

Atoms of cyclic edge connectivity in regular graphs

A cyclic edge-cut of a connected graph \(G\) is an edge set, the removal of which separates two cycles. If \(G\) has a cyclic edge-cut, then it is called cyclically separable. For a cyclically separable graph \(G\), the cyclic edge connectivity of a graph \(G\), denoted by \(\lambda _c(G)\), is the minimum cardinality over all cyclic edge cuts. Let \(X\) be a non-empty proper subset of \(V(G)\). If \([X,\overline{X}]=\{xy\in E(G)\ |\ x\in X, y\in \overline{X}\}\) is a minimum cyclic edge cut of \(G\), then \(X\) is called a \(\lambda _c\) -fragment of \(G\). A \(\lambda _c\)-fragment with minimum cardinality is called a \(\lambda _c\) -atom. Let \(G\) be a \(k (k\ge 3)\)-regular cyclically separable graph with \(\lambda _c(G)<g(k-2)\), where \(g\) is the girth of \(G\). A combination of the results of Nedela and Skoviera (Math Slovaca 45:481–499, 1995) and Xu and Liu (Australas J Combin 30:41–49, 2004) gives that if \(k\ne 5\) then any two distinct \(\lambda _c\)-atoms of \(G\) are disjoint. The remaining case of \(k=5\) is considered in this paper, and a new proof for Nedela and ?koviera’s result is also given. As a result, we obtain the following result. If \(X\) and \(X'\) are two distinct \(\lambda _c\)-atoms of \(G\) such that \(X\cap X'\ne \emptyset \), then \((k,g)=(5,3)\) and \(G[X]\cong K_4\). As corollaries, several previous results are easily obtained.     

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 \(\chi _{\prod }^{\prime \prime }(G)\le \Delta (G)+2\).     

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.