Choosability in signed planar graphs

This paper studies the choosability of signed planar graphs. We prove that every signed planar graph is 5-choosable and that there is a signed planar graph which is not 4-choosable while the unsigned graph is 4-choosable. For each $k \in \{3,4,5,6\}$, every signed planar graph without circuits of length $k$ is 4-choosable. Furthermore, every signed planar graph without circuits of length 3 and of length 4 is 3-choosable. We construct a signed planar graph with girth 4 which is not 3-choosable but the unsigned graph is 3-choosable.Comment: We updated the reference lis

A refinement of choosability of graphs

Assume $k$ is a positive integer, $\lambda=\{k_1, k_2, \ldots, k_q\}$ is a partition of $k$ and $G$ is a graph. A $\lambda$-list assignment of $G$ is a $k$-list assignment $L$ of $G$ such that the colour set $\cup_{v\in V(G)}L(v)$ can be partitioned into $q$ subsets $C_1 \cup C_2 \ldots \cup C_q$ and for each vertex $v$ of $G$, $|L(v) \cap C_i| \ge k_i$. We say $G$ is $\lambda$-choosable if for each $\lambda$-list assignment $L$ of $G$, $G$ is $L$-colourable. It follows from the definition that if $\lambda =\{k\}$, then $\lambda$-choosable is the same as $k$-choosable, if $\lambda =\{1,1,\ldots, 1\}$, then $\lambda$-choosable is equivalent to $k$-colourable. For the other partitions of $k$ sandwiched between $\{k\}$ and $\{1,1,\ldots, 1\}$ in terms of refinements, $\lambda$-choosability reveals a complex hierarchy of colourability of graphs. We prove that for two partitions $\lambda, \lambda'$ of $k$, every $\lambda$-choosable graph is $\lambda'$-choosable if and only if $\lambda'$ is a refinement of $\lambda$. Then we concentrate on $\lambda$-choosability of planar graphs for partitions $\lambda$ of $4$. Several conjectures concerning colouring of generalized signed planar graphs are proposed and relations between these conjectures and list colouring conjectures for planar graphs are explored. In particular, it is proved that a conjecture of K\"{u}ndgen and Ramamurthi on list colouring of planar graphs is implied by the conjecture that every planar graph is $\{2,2\}$-choosable, and also implied by the conjecture of M\'{a}\v{c}ajov\'{a}, Raspaud and \v{S}koviera which asserts that every planar graph is signed MRS-$4$-colourable, and that a conjecture of Kang and Steffen asserting that every planar graph is signed KS-$4$-colourable implies that every planar graph is $\{1,1,2\}$-choosable.Comment: 10 pages, 1 figur

A Sufficient condition for DP-4-colorability

DP-coloring of a simple graph is a generalization of list coloring, and also a generalization of signed coloring of signed graphs. It is known that for each $k \in \{3, 4, 5, 6\}$, every planar graph without $C_k$ is 4-choosable. Furthermore, Jin, Kang, and Steffen \cite{JKS} showed that for each $k \in \{3, 4, 5, 6\}$, every signed planar graph without $C_k$ is signed 4-choosable. In this paper, we show that for each $k \in \{3, 4, 5, 6\}$, every planar graph without $C_k$ is 4-DP-colorable, which is an extension of the above results

The Alon-Tarsi number of a planar graph minus a matching

This paper proves that every planar graph $G$ contains a matching $M$ such that the Alon-Tarsi number of $G-M$ is at most $4$. As a consequence, $G-M$ is $4$-paintable, and hence $G$ itself is $1$-defective $4$-paintable. This improves a result of Cushing and Kierstead [Planar Graphs are 1-relaxed, 4-choosable, {\em European Journal of Combinatorics} 31(2010),1385-1397], who proved that every planar graph is $1$-defective $4$-choosable.Comment: 11 page

List Colouring Big Graphs On-Line

In this paper, we investigate the problem of graph list colouring in the on-line setting. We provide several results on paintability of graphs in the model introduced by Schauz [13] and Zhu [20]. We prove that the on-line version of Ohba's conjecture is true in the class of planar graphs. We also consider several alternate on-line list colouring models

Planar graphs without 4-cycles adjacent to triangles are DP-4-colorable

DP-coloring (also known as correspondence coloring) of a simple graph is a generalization of list coloring. It is known that planar graphs without 4-cycles adjacent to triangles are 4-choosable, and planar graphs without 4-cycles are DP-4-colorable. In this paper, we show that planar graphs without 4-cycles adjacent to triangles are DP-4-colorable, which is an extension of the two results above.Comment: 15 pages, 5 figure

On two generalizations of the Alon-Tarsi polynomial method

In a seminal paper, Alon and Tarsi have introduced an algebraic technique for proving upper bounds on the choice number of graphs (and thus, in particular, upper bounds on their chromatic number). The upper bound on the choice number of $G$ obtained via their method, was later coined the \emph{Alon-Tarsi number of $G$} and was denoted by $AT(G)$. They have provided a combinatorial interpretation of this parameter in terms of the eulerian subdigraphs of an appropriate orientation of $G$. Their characterization can be restated as follows. Let $D$ be an orientation of $G$. Assign a weight $\omega_D(H)$ to every subdigraph $H$ of $D$: if $H \subseteq D$ is eulerian, then $\omega_D(H) = (-1)^{e(H)}$, otherwise $\omega_D(H) = 0$. Alon and Tarsi proved that $AT(G) \leq k$ if and only if there exists an orientation $D$ of $G$ in which the out-degree of every vertex is strictly less than $k$, and moreover $\sum_{H \subseteq D} \omega_D(H) \neq 0$. Shortly afterwards, for the special case of line graphs of $d$-regular $d$-edge-colorable graphs, Alon gave another interpretation of $AT(G)$, this time in terms of the signed $d$-colorings of the line graph. In this paper we generalize both results. The first characterization is generalized by showing that there is an infinite family of weight functions (which includes the one considered by Alon and Tarsi), each of which can be used to characterize $AT(G)$. The second characterization is generalized to all graphs (in fact the result is even more general -- in particular it applies to hypergraphs). We then use the second generalization to prove that $\chi(G) = ch(G) = AT(G)$ holds for certain families of graphs $G$. Some of these results generalize certain known choosability results

Degree choosable signed graphs

A signed graph is a graph in which each edge is labeled with $+1$ or $-1$. A (proper) vertex coloring of a signed graph is a mapping \f that assigns to each vertex $v\in V(G)$ a color \f(v)\in \mz such that every edge $vw$ of $G$ satisfies \f(v)\not= \sg(vw)\f(w), where \sg(vw) is the sign of the edge $vw$. For an integer $h\geq 0$, let \Ga_{2h}=\{\pm1,\pm2, \ldots, \pm h\} and \Ga_{2h+1}=\Ga_{2h} \cup \{0\}. Following \cite{MaRS2015}, the signed chromatic number \scn(G) of $G$ is the least integer $k$ such that $G$ admits a vertex coloring \f with {\rm im}(\f)\subseteq \Ga_k. As proved in \cite{MaRS2015}, every signed graph $G$ satisfies \scn(G)\leq \De(G)+1 and there are three types of signed connected simple graphs for which equality holds. We will extend this Brooks' type result by considering graphs having multiple edges. We will also proof a list version of this result by characterizing degree choosable signed graphs. Furthermore, we will establish some basic facts about color critical signed graphs.Comment: 21 page

Concepts of signed graph coloring

This paper surveys recent development of concepts related to coloring of signed graphs. Various approaches are presented and discussed.Comment: revised version, 28 page

Alon-Tarsi number of signed planar graphs

Let $(G,\sigma)$ be any signed planar graph. We show that the Alon-Tarsi number of $(G,\sigma)$ is at most 5, generalizing a recent result of Zhu for unsigned case. In addition, if $(G,\sigma)$ is $2$-colorable then $(G,\sigma)$ has the Alon-Tarsi number at most 4. We also construct a signed planar graph which is $2$-colorable but not $3$-choosable.Comment: 14 pages,2 figure