M‐degrees of quadrangle‐free planar graphs

This article proves the following result: Let G and G′ be graphs of orders n and n′, respectively. Let G* be obtained from G by adding to each vertex a set of n′ degree 1 neighbors. If G* has game coloring number m and G′ has acyclic chromatic number k, then the Cartesian product G□G′ has game chromatic number at most k(k + m − 1). As a consequence, the Cartesian product of two forests has game chromatic number at most 10, and the Cartesian product of two planar graphs has game chromatic number at most 105. © 2008 Wiley Periodicals, Inc. J Graph Theory 59: 261–278, 2008

Wagner's theorem for Torus graphs

The coloring game on planar graphs with large girth, by a result on sparse cactuses

For a real number $r\ge 2$, a circular $r$-colouring of a signed graph $(G, \sigma)$ is a mapping $c: V(G)\to [0, r)$ such that $|c(x)-c(y)|\in [1,r-1]$ for each positive edge $xy$ and $|c(x)-c(y)|\in [0,r/2-1]\cup [r/2+1,r)$ for each negative edge $xy$. This concept is recently introduced by Naserasr, Wang, and Zhu in 2021, and they show that for any $\varepsilon>0$, there exist signed planar bipartite graphs (of girth 4) which are not circular $(4-\varepsilon)$-colourable. In this paper, we prove that for each signed planar graph $(G, \sigma)$ of girth at least $5$, there exists a real number $\varepsilon=\varepsilon(G,\sigma)>0$ such that $(G, \sigma)$ is circular $(4-\varepsilon)$-colorable. Our proof utilizes a Thomassen-type inductive argument on the dual version in terms of circular flows, which is motivated by a result of Richter, Thomassen, and Younger (2016) on group connectivity of $5$-edge-connected planar graphs.

It is known that not all planar graphs are 4‐choosable; neither all of them are vertex 2‐arborable. However, planar graphs without 4‐cycles and even those without 4‐cycles adjacent to 3‐cycles are known to be 4‐choosable. We extend this last result in terms of covering the vertices of a graph by induced subgraphs of variable degeneracy. In particular, we prove that every planar graph without 4‐cycles adjacent to 3‐cycles can be covered by two induced forests. © 2009 Wiley Periodicals, Inc. J Graph Theory 62, 234–240, 2009

A plane graph G is coupled k‐choosable if, for any list assignment L satisfying $|{{L}}({{x}})|= {{k}}$ for every ${{x}}\in {{V}}({{G}})\cup {{F}}({{G}})$, there is a coloring that assigns to each vertex and each face a color from its list such that any two adjacent or incident elements receive distinct colors. We prove that every plane graph is coupled 7‐choosable. We further show that maximal plane graphs, ${{K}}_{{4}}$‐minor free graphs, and plane graphs with maximum degree at most three are coupled 6‐choosable. © 2008 Wiley Periodicals, Inc. J Graph Theory 58: 27–44, 2008

We consider the following problem which is motivated by two different contexts independently, namely graph theory and combinatorial optimization.Given a 3-connected planar graph G with n vertices, is there a spanning closed walk W with at most 4n/3 edges?In graph theory, the above question is motivated by the famous hamiltonian result by Tutte in 1956 which says that every 4-connected planar graph is hamiltonian (a simpler proof is given by Thomassen in 1983). What happens if we relax the 4-connectivity? There is a 3-connected planar graph that is not hamiltonian, but how about a spanning close walk (which is exactly a traveling salesman tour. Sometimes such a walk is called hamiltonian walk)? How many edges are necessary to cover all the vertices of a 3-connected planar graph by a closed walk? This is exactly the above question.In combinatorial optimization, the famous traveling salesman problem in metric graphs is one of most fundamental NP-hard optimization problems. In spite of a vast amount of research several important questions remain open. In particular, the best known upper bound is not believed to be best possible. A promising direction to improve this approximation guarantee, has long been to understand the power of a linear program known as the Held-Karp relaxation [11]. On the one hand, the best lower bound on its integrality gap (for the symmetric case) is 4/3 and indeed the famous (so called 4/3-)conjecture said that this lower bound would be tight [10]. Goemans pointed out that there is a planar graph that achieves this bound. So he brought attention to the above question, i.e, the famous 4/3-conjecture is always true for 3-connected planar graphs.We prove the above problem in the following strong form;Given a circuit graph (which is obtained from a 3-connected planar graph by deleting one vertex) with n vertices, there is a spanning closed walk with at most 4(n − 1)/3 edges such that each edge is used at most twice.Moreover, our proof is constructive (and purely combinatorial) in a sense that there is an O(n2) algorithm to construct, given a 3-connected planar graph, such a walk.We shall construct an example that shows that the bound 4(n − 1)/3 is essentially tight. We also point out that 2-connected planar graphs may not have such a walk, as K2,n−2 shows.

Deciding whether a grid is a topological subgraph of a planar graph is NP-complete

While the game chromatic number of a forest is known to be at most 4, no simple criteria are known for determining the game chromatic number of a forest. We first state necessary and sufficient conditions for forests with game chromatic number 2 and then investigate the differences between forests with game chromatic number 3 and 4. In doing so, we present a minimal example of a forest with game chromatic number 4, criteria for determining in polynomial time the game chromatic number of a forest without vertices of degree 3, and an example of a forest with maximum degree 3 and game chromatic number 4. This gives partial progress on the open question of the computational complexity of the game chromatic number of a forest.

Sufficient conditions for a planar graph to be list edge [formula omitted]-colorable and list totally [formula omitted]-colorable

The grid theorem, originally proved by Robertson and Seymour in 1986 [RS10, Graph Minors V], is one of the most central results in the study of graph minors and has found many algorithmic applications, especially in the analysis of routing problems. The relation between and grid minors is particularly tight for planar graphs, as every planar graph of at least 6k contains a grid of order k as a minor [RST94]. This polynomial, in fact linear, bound on the size of grid minors has enabled many important consequences, such as sublinear separators and subexponential algorithms for many NP-hard problems on planar graphs. In the mid-90s, Reed and Johnson, Robertson, Seymour and Thomas proposed a notion of directed and conjectured an excluded grid theorem for directed graphs. This theorem was proved in 2015 [KK15] by the latter two authors but the function relating directed and grid minors is very big, even in the planar case.Directed grids have found several algorithmic applications such as low-congestion routing. See e.g. [CE15, CEP16, KKK14, EMW16, AKKW16]. However, in the undirected case the polynomial, in fact linear, bound on the size of grid minors in planar graphs have made this tool so extremely successful. Consequently, the lack of polynomial bounds for directed grid minors in planar digraphs has so far prevented further applications of this technique in the directed setting. The main result of this paper is to close this gap and to establish a polynomial bound for the directed grid theorem on planar digraphs. We are optimistic that this will enable further applications of directed and its dual notion of directed grids in the context of planar digraphs.Towards the end, we also give treewidth sparsifier for directed graphs, which has been already considered in undirected graphs. This result allows us to obtain an Eulerian subgraph of bounded degree in D that still has high directed treewidth. We believe this result is of independent interest for structural graph theory.

We consider the degree/diameter problem for graphs embedded in a surface, namely, given a surface $\Sigma$ and integers $\Delta$ and $k$, determine the maximum order $N(\Delta,k,\Sigma)$ of a graph embeddable in $\Sigma$ with maximum degree $\Delta$ and diameter $k$. We introduce a number of constructions which produce many new largest known planar and toroidal graphs. We record all these graphs in the available tables of largest known graphs. Given a surface $\Sigma$ of Euler genus $g$ and an odd diameter $k$, the current best asymptotic lower bound for $N(\Delta,k,\Sigma)$ is given by \[\sqrt{\frac{3}{8}g}\Delta^{\lfloor k/2\rfloor}.\] Our constructions produce new graphs of order \[\begin{cases}6\Delta^{\lfloor k/2\rfloor}& \text{if $\Sigma$ is the Klein bottle}\\ \(\frac{7}{2}+\sqrt{6g+\frac{1}{4}}\)\Delta^{\lfloor k/2\rfloor}& \text{otherwise,}\end{cases}\] thus improving the former value by a factor of 4.

Several games that arise from graph coloring have been introduced and studied. Let \(\varphi \) denote a graph invariant that arises from such a game. If G is a graph and \(\varphi (G-x)\ne \varphi (G)=k\), \(k \ge 1\), holds true for every vertex \(x \in V(G)\), then G is called a k-\(\varphi \)-game-vertex-critical graph. We study the concept of \(\varphi \)-game-vertex-criticality for \(\varphi \in \{\chi _g, \chi _i, \chi _\mathrm{{ig}}^\mathrm{{A}}, \chi _\mathrm{{ig}}^\mathrm{{AB}}\}\), where \(\chi _g\) denotes the standard game chromatic number, \(\chi _i\) denotes the indicated game chromatic number and \(\chi _\mathrm{{ig}}^\mathrm{{A}}\), \(\chi _\mathrm{{ig}}^\mathrm{{AB}}\) denote two versions of the independence game chromatic number. Since the game chromatic number \(\varphi (G-x)\) can either decrease or increase with respect to \(\varphi (G)\), we distinguish between lower, upper and mixed vertex-criticality. We show that for \(\varphi \in \{\chi _g, \chi _\mathrm{{ig}}^\mathrm{{A}}, \chi _\mathrm{{ig}}^\mathrm{{AB}}\}\) the difference \(\varphi (G)-\varphi (G-x)\), \(x \in V(G)\), can be arbitrarily large. A characterization of 2-\(\varphi \)-game-vertex-critical and (connected) 3-\(\varphi \)-lower-game-vertex-critical graphs for all \(\varphi \in \{\chi _g, \chi _i, \chi _\mathrm{{ig}}^\mathrm{{A}}, \chi _\mathrm{{ig}}^\mathrm{{AB}}\}\) is given. It is shown that \(\chi _g\)-game-vertex-critical, \(\chi _\mathrm{{ig}}^\mathrm{{A}}\)-game-vertex-critical and \(\chi _\mathrm{{ig}}^\mathrm{{AB}}\)-game-vertex-critical graphs are not necessarily connected. However, it is also shown that \(\chi _i\)-lower-game-vertex-critical graphs are always connected.

Given two points s and t in the plane and a set of obstacles defined by closed curves, what is the minimum number of obstacles touched by a path connecting s and t? This is a fundamental and well-studied problem arising naturally in computational geometry, graph theory (under the names Min-Color Path and Minimum Label Path), wireless sensor networks (Barrier Resilience) and motion planning (Minimum Constraint Removal). It remains NP-hard even for very simple-shaped obstacles such as unit-length line segments. In this paper we give the first constant factor approximation algorithm for this problem, resolving an open problem of [Chan and Kirkpatrick, TCS, 2014] and [Bandyapadhyay et al., CGTA, 2020]. We also obtain a constant factor approximation for the Minimum Color Prize Collecting Steiner Forest where the goal is to connect multiple request pairs (s1, t1), . . . ,(sk, tk) while minimizing the number of obstacles touched by any (si, ti) path plus a fixed cost of wi for each pair (si, ti) left disconnected. This generalizes the classic Steiner Forest and Prize-Collecting Steiner Forest problems on planar graphs, for which intricate PTASes are known. In contrast, no PTAS is possible for Min-Color Path even on planar graphs since the problem is known to be APXhard [Eiben and Kanj, TALG, 2020]. Additionally, we show that generalizations of the problem to disconnected obstacles

1-planar graphs are odd 13-colorable