Chromatic Number Of Planar Graphs Research Articles

On the weak 2-coloring number of planar graphs

For a graph G=(V,E), a total ordering L on V, and a vertex v∈V, let Wcol2[G,L,v] be the set of vertices w∈V for which there is a path from v to w whose length is 0, 1 or 2 and whose L-least vertex is w. The weak 2-coloring number wcol2(G) of G is the least k such that there is a total ordering L on V with |Wcol2[G,L,v]|≤k for all vertices v∈V. We improve the known upper bound on the weak 2-coloring number of planar graphs from 28 to 23. As the weak 2-coloring number is the best known upper bound on the star list chromatic number of planar graphs, this bound is also improved.

Planar graphs have bounded nonrepetitive chromatic number

The following seemingly simple question with surprisingly many connections to various problems in computer science and mathematics can be traced back to the beginning of the 20th century to the work of [Axel Thue](https://en.wikipedia.org/wiki/Axel_Thue): How many colors are needed to color the positive integers in a way such that no two consecutive segments of the same length have the same color pattern? Clearly, at least three colors are needed: if there was such a coloring with two colors, then any two consecutive integers would have different colors (otherwise, we would get two consecutive segments of length one with the same color pattern) and so the colors would have to alternate, i.e., any two consecutive segments of length two would have the same color pattern. Suprisingly, three colors suffice. The coloring can be constructed as follows. We first define a sequence of 0s and 1s recursively as follows: we start with 0 only and in each step we take the already constructed sequence, flip the 0s and 1s in it and append the resulting sequence at the end. In this way, we sequentially obtain the sequences 0, 01, 0110, 01101001, etc., which are all extensions of each other. The limiting infinite sequence is known as the [Thue-Morse sequence](https://en.wikipedia.org/wiki/Thue%E2%80%93Morse_sequence). Another view of the sequence is that the $i$-th element is the parity of the number of 1s in the binary representation of $i-1$, i.e., it is one if the number is odd and zero if it is even. The coloring of integers is obtained by coloring an integer $i$ by the difference of the $(i+1)$-th and $i$-th entries in the Thue-Morse sequence, i.e., the sequence of colors will be 1, 0, -1, 1, -1, 0, 1, 0, etc. One of the properties of the Thue-Morse sequence is that it does not containing two overlapping squares, i.e., there is no sequence X such that 0X0X0 or 1X1X1 would be a subsequence of the Thue-Morse sequence. This implies that the coloring of integers that we have constructed has no two consecutive segments with the same color pattern. The article deals with a generalization of this notion to graphs. The _nonrepetitive chromatic number_ of a graph $G$ is the minimum number of colors required to color the vertices of $G$ in such way that no path with an even number of vertices is comprised of two paths with the same color pattern. The construction presented above yields that the nonrepetitive chromatic number of every path with at least four vertices is three. The article answers in the positive the following question of Alon, Grytczuk, Hałuszczak and Riordan from 2002: Is the nonrepetitive chromatic number of planar graphs bounded? They show that the nonrepetitive chromatic number of every planar graph is at most 768 and provide generalizations to graphs embeddable to surfaces of higher genera and more generally to classes of graphs excluding a (topological) minor. Before their work, the best upper bound on the nonrepetitive chromatic number of planar graphs was logarithmic in their number of vertices, in addition to a universal upper bound quadratic in the maximum degree of a graph obtained using probabilistic method. The key ingredient for the argument presented in the article is the recent powerful result by Dujmović, Joret, Micek, Morin, Ueckerdt and Wood asserting that every planar graph is a subgraph of the strong product of a path and a graph of bounded tree-width (tree-shaped graph).

Neighbor sum distinguishing total chromatic number of planar graphs without 5-cycles

Neighbor sum distinguishing total chromatic number of planar graphs without 5-cycles

A characterization for the neighbor-distinguishing total chromatic number of planar graphs with [formula omitted

The neighbor-distinguishing total chromatic number χa′′(G) of a graph G is the smallest integer k such that G can be totally colored using k colors with a condition that any two adjacent vertices have different sets of colors. In this paper, we give a sufficient and necessary condition for a planar graph G with maximum degree 13 to have χa′′(G)=14 or χa′′(G)=15. Precisely, we show that if G is a planar graph of maximum degree 13, then 14≤χa′′(G)≤15; and χa′′(G)=15 if and only if G contains two adjacent 13-vertices.

Neighbor sum distinguishing total chromatic number of planar graphs

Let G = (V(G), E(G)) be a graph and ϕ be a proper k-total coloring of G. Set fϕ(v)=∑uv∈E(G)ϕ(uv)+ϕ(v), for each v ∈ V(G). If fϕ(u) ≠ fϕ(v) for each edge uv ∈ E(G), the coloring ϕ is called a k-neighbor sum distinguishing total coloring of G. The smallest integer k in such a coloring of G is the neighbor sum distinguishing total chromatic number, denoted by χΣ″(G). In this paper, by using the famous Combinatorial Nullstellensatz, we determine χΣ″(G) for any planar graph G with Δ(G) ≥ 13.

On $t$-Common List-Colorings

In this paper, we introduce a new variation of list-colorings. For a graph $G$ and for a given nonnegative integer $t$, a $t$-common list assignment of $G$ is a mapping $L$ which assigns each vertex $v$ a set $L(v)$ of colors such that given set of $t$ colors belong to $L(v)$ for every $v\in V(G)$. The $t$-common list chromatic number of $G$ denoted by $ch_t(G)$ is defined as the minimum positive integer $k$ such that there exists an $L$-coloring of $G$ for every $t$-common list assignment $L$ of $G$, satisfying $|L(v)| \ge k$ for every vertex $v\in V(G)$. We show that for all positive integers $k, \ell$ with $2 \le k \le \ell$ and for any positive integers $i_1 , i_2, \ldots, i_{k-2}$ with $k \le i_{k-2} \le \cdots \le i_1 \le \ell$, there exists a graph $G$ such that $\chi(G)= k$, $ch(G) = \ell$ and $ch_t(G) = i_t$ for every $t=1, \ldots, k-2$. Moreover, we consider the $t$-common list chromatic number of planar graphs. From the four color theorem and the result of Thomassen (1994), for any $t=1$ or $2$, the sharp upper bound of $t$-common list chromatic number of planar graphs is $4$ or $5$. Our first step on $t$-common list chromatic number of planar graphs is to find such a sharp upper bound. By constructing a planar graph $G$ such that $ch_1(G) =5$, we show that the sharp upper bound for $1$-common list chromatic number of planar graphs is $5$. The sharp upper bound of $2$-common list chromatic number of planar graphs is still open. We also suggest several questions related to $t$-common list chromatic number of planar graphs.

Neighbor sum distinguishing total chromatic number of planar graphs with maximum degree 10

Given a simple graph G, a proper total-k-coloring ϕ:V(G)∪E(G)→{1,2,…,k} is called neighbor sum distinguishing if Sϕ(u) ≠ Sϕ(v) for any two adjacent vertices u, v ∈ V(G), where Sϕ(u) is the sum of the color of u and the colors of the edges incident with u. It has been conjectured by Pilśniak and Woźniak that Δ(G)+3 colors enable the existence of a neighbor sum distinguishing total coloring. The conjecture is confirmed for any graph with maximum degree at most 3 and for planar graph with maximum degree at least 11. We prove that the conjecture holds for any planar graph G with Δ(G)=10. Moreover, for any planar graph G with Δ(G) ≥ 11, Δ(G)+2 colors guarantee such a total coloring, and the upper bound Δ(G)+2 is tight.

Proof of Four Color Map Theorem by Using PRN of Graph

This paper intends to study the relation between PRN and chromatic number of planar graphs. In this regard we investigate that isomorphic or 1 isomorphic graph may or may not have equal PRN and few other related results. Precisely, we give simple proof of Four Color Map Theorem.

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.

Adapted game colouring of graphs

Suppose G=(V,E) is a graph and F is a colouring of its edges (not necessarily proper) that uses the colour set X. In an adapted colouring game, Alice and Bob alternately colour uncoloured vertices of G with colours from X. A partial colouring c of the vertices of G is legal if there is no edge e=xy such that c(x)=c(y)=F(e). In the process of the game, each partial colouring must be legal. If eventually all the vertices of G are coloured, then Alice wins the game. Otherwise, Bob wins the game. The adapted game chromatic number of a graph G, denoted by χadg(G), is the minimum number of colours needed to ensure that for any edge colouring F of G, Alice has a winning strategy for the game. This paper studies the adapted game chromatic number of various classes of graphs. We prove that the maximum adapted game chromatic number of trees is 3, the maximum adapted game chromatic number of outerplanar graphs is 5, the adapted game chromatic number of partial k-trees is at most 2k+1, and the adapted game chromatic number of planar graphs is at most 11.

Chromatic number of sparse colored mixed planar graphs

A colored mixed graph has vertices linked by both colored arcs and colored edges. The chromatic number of such a graph G is defined as the smallest order of a colored mixed graph H such that there exists a (arc-color preserving) homomorphism from G to H. We study in this paper the colored mixed chromatic number of planar graphs, partial 2-trees and outerplanar graphs with given girth.

MINIMAX DEGREES OF QUASIPLANAR GRAPHS WITH NO SHORT CYCLES OTHER THAN TRIANGLES

For an edge $xy$, let $M(xy)$ be the maximum of the degrees of $x$ and $y$. The {\em minimax degree} (or $M$-degree) of a graph $G$ is $M^*(G)=\min\{M(xy)| xy\in E(G)\}$. In order to get upper bounds on the game chromatic number of planar graphs, He, Hou, Lih, Shao, Wang, and Zhu showed that every planar graph $G$ without leaves and $4$-cycles has minimax degree at most $8$, which was improved by Borodin, Kostochka, Sheikh, and Yu to the sharp bound $7$. We show that every planar graph $G$ without leaves and $4$- and $5$-cycles has $M$-degree at most $5$, which bound is sharp. We also show that every planar graph $G$ without leaves and cycles of length from $4$ to $7$ has $M$-degree at most $4$, which bound is attained even on planar graphs with no cycles of length from $4$ to arbitrarily large number. Besides, we give sufficient conditions for a planar graph to have $M$-degrees $3$ and $2$. Similar results are obtained for graphs embeddable into the projective plane, the torus and the Klein bottle.

Oriented colorings of partial 2-trees

A homomorphism from an oriented graph G to an oriented graph H is an arc-preserving mapping f from V ( G ) to V ( H ) , that is f ( x ) f ( y ) is an arc in H whenever xy is an arc in G. The oriented chromatic number of G is the minimum order of an oriented graph H such that G has a homomorphism to H. In this paper, we determine the oriented chromatic number of the class of partial 2-trees for every girth g ⩾ 3 . We also give an upper bound for the oriented chromatic number of planar graphs with girth at least 11.

Strong oriented chromatic number of planar graphs without cycles of specific lengths

Abstract A strong oriented k-coloring of an oriented graph G is a homomorphism ϕ from G to H having k vertices labelled by the k elements of an abelian additive group M, such that for any pairs of arcs u v → and z t → of G, we have ϕ ( v ) − ϕ ( u ) ≠ − ( ϕ ( t ) − ϕ ( z ) ) . The strong oriented chromatic number χ s ( G ) is the smallest k such that G admits a strong oriented k-coloring. In this paper, we consider the following problem: Let i ⩾ 4 be an integer. Let G be an oriented planar graph without cycles of lengths 4 to i. What is the strong oriented chromatic number of G?

Strong Oriented Chromatic Number of Planar Graphs without Short Cycles

Let M be an additive abelian group. An M-strong-oriented coloring of an oriented graph G is a mapping f from V(G) to M such that f(u) <> j(v) whenever uv is an arc in G and f(v)−f(u) <> −(f(t)−f(z)) whenever uv and zt are two arcs in G. The strong oriented chromatic number of an oriented graph is the minimal order of a group M such that G has an M-strong-oriented coloring. This notion was introduced by Nesetril and Raspaud [Ann. Inst. Fourier, 49(3):1037-1056, 1999]. We prove that the strong oriented chromatic number of oriented planar graphs without cycles of lengths 4 to 12 (resp. 4 or 6) is at most 7 (resp. 19). Moreover, for all i ≥ 4, we construct outerplanar graphs without cycles of lengths 4 to i whose oriented chromatic number is 7.

On total chromatic number of planar graphs without 4-cycles

Let G be a simple graph with maximum degree Δ(G) and total chromatic number x ve (G). Vizing conjectured that Δ(G) + 1 ⩽ X ve (G) ⩽ δ(G) + 2 (Total Chromatic Conjecture). Even for planar graphs, this conjecture has not been settled yet. The unsettled difficult case for planar graphs is Δ(G) = 6. This paper shows that if G is a simple planar graph with maximum degree 6 and without 4-cycles, then x ve (G) ⩽ 8. Together with the previous results on this topic, this shows that every simple planar graph without 4-cycles satisfies the Total Chromatic Conjecture.

Total chromatic number of planar graphs with maximum degree ten

AbstractIn this article we prove that the total chromatic number of a planar graph with maximum degree 10 is 11. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 91–102, 2007

Group chromatic number of planar graphs of girth at least 4

AbstractJeager et al. introduced a concept of group connectivity as a generalization of nowhere zero flows and its dual concept group coloring, and conjectured that every 5‐edge connected graph is Z3‐connected. For planar graphs, this is equivalent to that every planar graph with girth at least 5 must have group chromatic number at most 3. In this article, we show that if G is a plane graph with girth at least 4 such that all 4 cycles are independent, every 4‐cycle is a facial cycle and the distance between every pair of a 4‐cycle and a 5‐cycle is at least 1, then the group chromatic number of G is at most 3. As a special case, we show that the conjecture above holds for planar graphs. We also prove that if G is a connected K3,3‐minor free graph with girth at least 5, then the group chromatic number is at most 3. © 2006 Wiley Periodicals, Inc. J Graph Theory 52: 51–72, 2006

A note on group colorings

AbstractWe study a concept of group coloring introduced by Jaeger et al. We show that the group chromatic number of a graph with minimum degree δ is greater than δ/(2, ln δ) and we answer several open questions on the group chromatic number of planar graphs: a construction of a bipartite planar graph with group chromatic number four and a 3‐colorable planar graph with group chromatic number five are presented. We also observe that several upper bounds on the choice number for various subclasses of planar graphs also translate to the concept of group colorings. © 2005 Wiley Periodicals, Inc. J Graph Theory

Girth and fractional chromatic number of planar graphs

In 1959, even before the Four-Color Theorem was proved, Grötzsch showed that planar graphs with girth at least 4 have chromatic number at the most 3. We examine the fractional analogue of this theorem and its generalizations. For any fixed girth, we ask for the largest possible fractional chromatic number of a planar graph with that girth, and we provide upper and lower bounds for this quantity. © 2002 Wiley Periodicals, Inc. J Graph Theory 39: 201–217, 2002; DOI 10.1002/jgt10024