Minimum choosability of planar graphs

The problem of list coloring of graphs appears in practical problems concerning channel or frequency assignment. In this paper, we study the minimum number of choosability of planar graphs. A graph G is edge-k-choosable if whenever every edge x is assigned with a list of at least k colors, L(x)), there exists an edge coloring \(\phi \) such that \(\phi (x) \in L(x)\). Similarly, A graph G is toal-k-choosable if when every element (edge or vertex) x is assigned with a list of at least k colors, L(x), there exists an total coloring \(\phi \) such that \(\phi (x) \in L(x)\). We proved \(\chi '_{l}(G)=\Delta \) and \(\chi ''_{l}(G)=\Delta +1\) for a planar graph G with maximum degree \(\Delta \ge 8\) and without chordal 6-cycles, where the list edge chromatic number \(\chi '_{l}(G)\) of G is the smallest integer k such that G is edge-k-choosable and the list total chromatic number \(\chi ''_{l}(G)\) of G is the smallest integer k such that G is total-k-choosable.     

An optimal square coloring of planar graphs

The square coloring of a graph is to color the vertices of a graph at distance at most 2 with different colors. In 1977, Wegner posed a conjecture on square coloring of planar graphs. The conjecture is still open. In this paper, we prove that Wegner’s conjecture is true for planar graphs with girth at least?6.     

Every planar graph with cycles of length neither 4 nor 5 is (1,1,0)-colorable

Let \(d_1, d_2,\dots ,d_k\) be \(k\) non-negative integers. A graph \(G\) is \((d_1,d_2,\ldots ,d_k)\) -colorable, if the vertex set of \(G\) can be partitioned into subsets \(V_1,V_2,\ldots ,V_k\) such that the subgraph \(G[V_i]\) induced by \(V_i\) has maximum degree at most \(d_i\) for \(i=1,2,\ldots ,k.\) Let \(\digamma \) be the family of planar graphs with cycles of length neither 4 nor 5. Steinberg conjectured that every graph of \(\digamma \) is \((0,0,0)\) -colorable. In this paper, we prove that every graph of \(\digamma \) is \((1,1,0)\) -colorable.     

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.     

Total coloring of planar graphs without short cycles

The total chromatic number of a graph \(G\), denoted by \(\chi ''(G)\), is the minimum number of colors needed to color the vertices and edges of \(G\) such that no two adjacent or incident elements get the same color. It is known that if a planar graph \(G\) has maximum degree \(\Delta (G)\ge 9\), then \(\chi ''(G)=\Delta (G)+1\). In this paper, it is proved that if \(G\) is a planar graph with \(\Delta (G)\ge 7\), and for each vertex \(v\), there is an integer \(k_v\in \{3,4,5,6,7,8\}\) such that there is no \(k_v\)-cycle which contains \(v\), then \(\chi ''(G)=\Delta (G)+1\).     

Star list chromatic number of planar subcubic graphs

A proper coloring of the vertices of a graph G is called a star-coloring if the union of every two color classes induces a star forest. The graph G is L-star-colorable if for a given list assignment L there is a star-coloring π such that π(v)∈L(v). If G is L-star-colorable for any list assignment L with |L(v)|≥k for all v∈V(G), then G is called k-star-choosable. The star list chromatic number of G, denoted by $\chi_{s}^{l}(G)$ , is the smallest integer k such that G is k-star-choosable. In this paper, we prove that every planar subcubic graph is 6-star-choosable.     

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 \(\varDelta \ge 27\).     

Acyclic edge coloring of planar graphs without a 3-cycle adjacent to a 6-cycle

An acyclic edge coloring of a graph \(G\) is a proper edge coloring such that no bichromatic cycles are produced. The acyclic chromatic index \(a'(G)\) of \(G\) is the smallest integer \(k\) such that \(G\) has an acyclic edge coloring using \(k\) colors. Fiam? ik (Math Slovaca 28:139–145, 1978) and later Alon et al. (J Graph Theory 37:157–167, 2001) conjectured that \(a'(G)\le \Delta +2\) for any simple graph \(G\) with maximum degree \(\Delta \) . In this paper, we confirm this conjecture for planar graphs without a \(3\) -cycle adjacent to a \(6\) -cycle.     

Acyclic edge coloring of planar graphs without 4-cycles

An acyclic edge coloring of a graph G is a proper edge coloring such that no bichromatic cycles are produced. The acyclic chromatic index a′(G) of G is the smallest integer k such that G has an acyclic edge coloring using k colors. Fiam?ik (Math. Slovaca 28:139–145, 1978) and later Alon, Sudakov and Zaks (J. Graph Theory 37:157–167, 2001) conjectured that a′(G)≤Δ+2 for any simple graph G with maximum degree Δ. In this paper, we confirm this conjecture for planar graphs G with Δ≠4 and without 4-cycles.     

Small grid drawings of planar graphs with balanced partition

In a grid drawing of a planar graph, every vertex is located at a grid point, and every edge is drawn as a straight-line segment without any edge-intersection. It is known that every planar graph G of n vertices has a grid drawing on an (n?2)×(n?2) or (4n/3)×(2n/3) integer grid. In this paper we show that if a planar graph G has a balanced partition then G has a grid drawing with small grid area. More precisely, if a separation pair bipartitions G into two edge-disjoint subgraphs G ₁ and?G ₂, then G has a max?{n ₁,n ₂}×max?{n ₁,n ₂} grid drawing, where n ₁ and n ₂ are the numbers of vertices in G ₁ and?G ₂, respectively. In particular, we show that every series-parallel graph G has a (2n/3)×(2n/3) grid drawing and a grid drawing with area smaller than 0.3941n ² (<(2/3)² n ²).     

The adjacent vertex distinguishing total coloring of planar graphs

An adjacent vertex distinguishing total coloring of a graph G is a proper total coloring of G such that any pair of adjacent vertices have distinct sets of colors. The minimum number of colors needed for an adjacent vertex distinguishing total coloring of G is denoted by $\chi''_{a}(G)$ . In this paper, we characterize completely the adjacent vertex distinguishing total chromatic number of planar graphs G with large maximum degree Δ by showing that if Δ≥14, then $\varDelta+1\leq \chi''_{a}(G)\leq \varDelta+2$ , and $\chi''_{a}(G)=\varDelta+2$ if and only if G contains two adjacent vertices of maximum degree.     

Total coloring of planar graphs without adjacent short cycles

In the study of computer science, optimization, computation of Hessians matrix, graph coloring is an important tool. In this paper, we consider a classical coloring, total coloring. Let \(G=(V,E)\) be a graph. Total coloring is a coloring of \(V\cup {E}\) such that no two adjacent or incident elements (vertex/edge) receive the same color. Let G be a planar graph with \(\varDelta \ge 8\). We proved that if for every vertex \(v\in V\), there exists two integers \(i_v,j_v\in \{3,4,5,6,7\}\) such that v is not incident with adjacent \(i_v\)-cycles and \(j_v\)-cycles, then the total chromatic number of graph G is \(\varDelta +1\).     

2-Distance coloring of planar graphs with girth 5

A vertex coloring is said to be 2-distance if any two distinct vertices of distance at most 2 receive different colors. Let G be a planar graph with girth at least 5. In this paper, we prove that G admits a 2-distance coloring with at most \(\Delta (G)+3\) colors if \(\Delta (G)\ge 339\).     

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 \(\chi _{2}(G)\le 5\Delta -7\).     

List-edge-coloring of planar graphs without 6-cycles with three chords

A graph G is edge-k-choosable if, whenever we are given a list L(e) of colors with \(|L(e)|\ge k\) for each \(e\in E(G)\), we can choose a color from L(e) for each edge e such that no two adjacent edges receive the same color. In this paper we prove that if G is a planar graph, and each 6-cycle contains at most two chords, then G is edge-k-choosable, where \(k=\max \{8,\Delta (G)+1\}\), and edge-t-choosable, where \(t=\max \{10,\Delta (G)\}\).     

Total coloring of planar graphs without adjacent chordal 6-cycles

A total coloring of a graph G is a coloring such that no two adjacent or incident elements receive the same color. In this field there is a famous conjecture, named Total Coloring Conjecture, saying that the the total chromatic number of each graph G is at most \(\Delta +2\). Let G be a planar graph with maximum degree \(\Delta \ge 7\) and without adjacent chordal 6-cycles, that is, two cycles of length 6 with chord do not share common edges. In this paper, it is proved that the total chromatic number of G is \(\Delta +1\), which partly confirmed Total Coloring Conjecture.     

The 2-surviving rate of planar graphs without 5-cycles

Let \(G\) be a connected graph with \(n\ge 2\) vertices. Let \(k\ge 1\) be an integer. Suppose that a fire breaks out at a vertex \(v\) of \(G\). A firefighter starts to protect vertices. At each step, the firefighter protects \(k\)-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 \(\hbox {sn}_k(v)\) denote the maximum number of vertices in \(G\) that the firefighter can save when a fire breaks out at vertex \(v\). The \(k\)-surviving rate \(\rho _k(G)\) of \(G\) is defined to be \(\frac{1}{n^2}\sum _{v\in V(G)} {\hbox {sn}}_{k}(v)\), which is the average proportion of saved vertices. In this paper, we prove that if \(G\) is a planar graph with \(n\ge 2\) vertices and without 5-cycles, then \(\rho _2(G)>\frac{1}{363}\).