Choosability Of Planar Graphs Research Articles

On [formula omitted]-choosability of planar graphs without short cycles

For a graph G and two positive integers t,r, a (t,t+r)-list assignment of G is a function L that assigns a set of permissible colors L(u) to every vertex u such that |L(u)|≥t and |L(u)∪L(w)|≥t+r when uw is an edge. The graph G is said to be (t,t+r)-choosable if G allows a proper coloring ψ satisfying ψ(u)∈L(u) for each u∈V(G) and each (t,t+r)-list assignment L of G. In this paper, we consider the (2,2+r)-choosability of planar graphs without short cycles. We show that: (1) if G is a planar graph contains no cycles of length 4, then G is (2,9)-choosable; (2) if G is a planar graph contains no cycles of lengths 4 and 5, then G is (2,7)-choosable.

A Note on Edge-Group Choosability of Planar Graphs without 5-Cycles

This paper is devoted to a study of the concept of edge-group choosability of graphs. We say that G is edge- k -group choosable if its line graph is k -group choosable. In this paper, we study an edge-group choosability version of Vizing conjecture for planar graphs without 5-cycles and for planar graphs without noninduced 5-cycles (2010 Mathematics Subject Classification: 05C15, 05C20).

Two results on K-(2,1)-total choosability of planar graphs

The [Formula: see text]-total choice number of [Formula: see text], denoted by [Formula: see text], is the minimum [Formula: see text] such that [Formula: see text] is [Formula: see text]-[Formula: see text]-total choosable. It was proved in [Y. Yu, X. Zhang and G. Z. Liu, List (d,1)-total labeling of graphs embedded in surfaces, Oper. Res. Trans. 15(3) (2011) 29–37.] that [Formula: see text] if [Formula: see text] is a graph embedded in surface with Euler characteristic [Formula: see text] and [Formula: see text] big enough. In this paper, we prove that: (i) if [Formula: see text] is a planar graph with [Formula: see text] and [Formula: see text]-cycle is not adjacent to [Formula: see text]-cycle, [Formula: see text], then [Formula: see text]; (ii) if [Formula: see text] is a planar graph with [Formula: see text] and [Formula: see text]-cycle is not adjacent to [Formula: see text]-cycle, where [Formula: see text], then [Formula: see text].

Path Choosability of Planar Graphs

A path coloring of a graph $G$ is a vertex coloring of $G$ such that each color class induces a disjoint union of paths. We consider a path-coloring version of list coloring for planar and outerplanar graphs. We show that if each vertex of a planar graph is assigned a list of $3$ colors, then the graph admits a path coloring in which each vertex receives a color from its list. We prove a similar result for outerplanar graphs and lists of size $2$.For outerplanar graphs we prove a multicoloring generalization. We assign each vertex of a graph a list of $q$ colors. We wish to color each vertex with $r$ colors from its list so that, for each color, the set of vertices receiving it induces a disjoint union of paths. We show that we can do this for all outerplanar graphs if and only if $q/r \ge 2$. For planar graphs we conjecture that a similar result holds with $q/r \ge 3$; we present partial results toward this conjecture.

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.

The adjacent vertex distinguishing total choosability of planar graphs with maximum degree at least eleven

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.

(4, 2)-Choosability of Planar Graphs with Forbidden Structures

All planar graphs are 4-colorable and 5-choosable, while some planar graphs are not 4-choosable. Determining which properties guarantee that a planar graph can be colored using lists of size four has received significant attention. In terms of constraining the structure of the graph, for any $$\ell \in \{3,4,5,6,7\}$$ , a planar graph is 4-choosable if it is $$\ell $$ -cycle-free. In terms of constraining the list assignment, one refinement of k-choosability is choosability with separation. A graph is (k, s)-choosable if the graph is colorable from lists of size k where adjacent vertices have at most s common colors in their lists. Every planar graph is (4, 1)-choosable, but there exist planar graphs that are not (4, 3)-choosable. It is an open question whether planar graphs are always (4, 2)-choosable. A chorded $$\ell $$ -cycle is an $$\ell $$ -cycle with one additional edge. We demonstrate for each $$\ell \in \{5,6,7\}$$ that a planar graph is (4, 2)-choosable if it does not contain chorded $$\ell $$ -cycles.

A note on the minimum number of choosability of planar graphs

The problem of minimum number of choosability of graphs was first introduced by Vizing. It appears in some practical problems when concerning frequency assignment. In this paper, we study two important list coloring, list edge coloring and list total coloring. We prove that $$\chi '_{l}(G)=\varDelta $$?l?(G)=Δ and $$\chi ''_{l}(G)=\varDelta +1$$?l??(G)=Δ+1 for planar graphs with $$\varDelta \ge 8$$Δ?8 and without adjacent 4-cycles.

Group edge choosability of planar graphs without adjacent short cycles

In this paper, we prove that 2-degenerate graphs and some planar graphs without adjacent short cycles are group (Δ(G)+1)-edge-choosable, and some planar graphs with large girth and maximum degree are group Δ(G)-edge-choosable.

(3, 1)*-choosability of planar graphs without adjacent short cycles

Abstract A list assignment of a graph G is a function L that assigns a list L(v) of colors to each vertex v ∈ V ( G ) . An ( L , d ) ⁎ -coloring is a mapping π that assigns a color π ( v ) ∈ L ( v ) to each vertex v ∈ V ( G ) so that at most d neighbors of v receive color π ( v ) . A graph G is said to be ( k , d ) ⁎ -choosable if it admits an ( L , d ) ⁎ -coloring for every list assignment L with | L ( v ) | ⩾ k for all v ∈ V ( G ) . In 2001, Lih et al. [Lih, K., Z. Song, W. Wang, and K. Zhang, A note on list improper coloring planar graphs, Appl. Math. Lett. 14 (2001), 269–273] proved that planar graphs without 4- and l-cycles are ( 3 , 1 ) ⁎ -choosable, where l ∈ { 5 , 6 , 7 } . Later, Dong and Xu [Dong, W., and B. Xu, A note on list improper coloring of plane graphs, Discrete Appl. Math. 157 (2009), 433–436] proved that planar graphs without 4- and l-cycles are ( 3 , 1 ) ⁎ -choosable, where l ∈ { 8 , 9 } . There exist planar graphs containing 4-cycles that are not ( 3 , 1 ) ⁎ -choosable (Crown, Crown and Woodall, 1986 [Cowen, L., R. Cowen, and D. Woodall, Defective colorings of graphs in surfaces: partitions into subgraphs of bounded valency, J. Graph Theory 10 (1986), 187–195]). This partly explains the fact that in all above known sufficient conditions for the ( 3 , 1 ) ⁎ -choosability of planar graphs the 4-cycles are completely forbidden. In this paper we allow 4-cycles nonadjacent to relatively short cycles. More precisely, we prove that every planar graph without 4-cycles adjacent to 3- and 4-cycles is ( 3 , 1 ) ⁎ -choosable. This is a common strengthening of all above mentioned results. Moreover as a consequence we give a partial answer to a question of Xu and Zhang [Xu, B., and H. Zhang, Every toroidal graph without adjacent triangles is ( 4 , 1 ) ⁎ -choosable, Discrete Appl. Math. 155 (2007), 74–78] and show that every planar graph without 4-cycles is ( 3 , 1 ) ⁎ -choosable.

Improper Choosability of Planar Graphs without 4-Cycles

Let $k\ge 2$ be a positive integer, and let $d$ be a nonnegative integer. A graph $G$ is $d$-improperly $k$-choosable, or simply, $(k,d)$-choosable if, for every list assignment $L$ with $|L(v)|\geq k$ for every $v\in V (G)$, there exists an $L$-coloring of $G$ such that each vertex of $G$ has at most $d$ neighbors colored the same color as itself. It is known that every planar graph with cycles of length neither 4 nor $k$ for some $k\in\{5,6,7,8,9\}$ is $(3,1)$-choosable. In this paper, we prove that every planar graph without cycles of length 4 is $(3,1)$-choosable. This is best possible in the following two senses: (i) there are planar graphs which are not $(3,1)$-colorable, hence not $(3,1)$-choosable; (ii) there are planar graphs without 4-cycles which are not $(3,0)$-colorable, hence not $(3,0)$-choosable.

Acyclic choosability of planar graphs: a Steinberg like approach

Abstract An acyclic coloring of a graph G is a coloring of its vertices such that : (i) no two adjacent vertices in G receive the same color and (ii) no bicolored cycles exist in G . A list assignment of G is a function L that assigns to each vertex v ∈ V ( G ) a list L ( v ) of available colors. Let G be a graph and L be a list assignment of G . The graph G is acyclically L -list colorable if there exists an acyclic coloring ϕ of G such that ϕ ( v ) ∈ L ( v ) for all v ∈ V ( G ) . If G is acyclically L -list colorable for any list assignment L with | L ( v ) | ⩾ k for all v ∈ V ( G ) , then G is acyclically k -choosable. In this note, we prove that every planar graph without cycles of lengths 4 to 12 is acyclically 3-choosable.

List colourings of planar graphs

A graph G = G ( V , E ) is called L- list colourable if there is a vertex colouring of G in which the colour assigned to a vertex v is chosen from a list L ( v ) associated with this vertex. We say G is k - choosable if all lists L ( v ) have the cardinality k and G is L-list colourable for all possible assignments of such lists. There are two classical conjectures from Erdős, Rubin and Taylor 1979 about the choosability of planar graphs: (1) every planar graph is 5-choosable and, (2) there are planar graphs which are not 4-choosable. We will prove the second conjecture.

Choosability of planar graphs

A graph G = G( V, E) with lists L( v), associated with its vertices v ∈ V, is called L-list colourable if there is a proper vertex colouring of G in which the colour assigned to a vertex v is chosen from L( v). We say G is k- choosable if there is at least one L-list colouring for every possible list assignment L with ∥ L( v)∥ = k ∀ v ∈ V( G). Now, let an arbitrary vertex v of G be coloured with an arbitrary colour f of L( v). We investigate whether the colouring of v can be continued to an L-list colouring of the whole graph. G is called free k-choosable if such an L-list colouring exists for every list assignment L (∥ L( v)∥ = k ∀ v ∈ V( G)), every vertex v and every colour f ∈ L( v). We prove the equivalence of the well-known conjecture of Erdős et al. (1979): “Every planar graph is 5-choosable” with the following conjecture: “Every planar graph is free 5-choosable”.

List colourings of planar graphs

A graph G = G( V, E) is called L- list colourable if there is a vertex colouring of G in which the colour assigned to a vertex v is chosen from a list L( v) associated with this vertex. We say G is k-choosable if all lists L( v) have the cardinality k and G is L-list colourable for all possible assignments of such lists. There are two classical conjectures from Erdős, Rubin and Taylor 1979 about the choosability of planar graphs: (1) every planar graph is 5-choosable and, (2) there are planar graphs which are not 4-choosable. We will prove the second conjecture.