Constructions of Large Graphs on Surfaces

The M‐degree of an edge xy in a graph is the maximum of the degrees of x and y. The M‐degree of a graph G is the minimum over M‐degrees of its edges. In order to get upper bounds on the game chromatic number, He et al showed that every planar graph G without leaves and 4‐cycles has M‐degree at most 8 and gave an example of such a graph with M‐degree 3. This yields upper bounds on the game chromatic number of C4‐free planar graphs. We determine the maximum possible M‐degrees for planar, projective‐planar and toroidal graphs without leaves and 4‐cycles. In particular, for planar and projective‐planar graphs this maximum is 7. © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 80–85, 2009

Wagner's theorem for Torus graphs

Equitable colorings of planar graphs with maximum degree at least nine

A graph G has a (d,h)-decomposition if there is a pair (D,F) such that F is a subgraph of G and D is an acyclic orientation of G−E(F), where the maximum degree of F is no more than h and the maximum out-degree of D is no more than d. This paper proves that toroidal graphs having no adjacent triangles are (3,1)-decomposable, and for {i,j}⊆{3,4,6}, toroidal graphs without i- and j-cycles are (2,1)-decomposable. As consequences of these results, toroidal graphs without adjacent triangles are 1-defective DP-4-colorable, and toroidal graphs without i- and j-cycles are 1-defective DP-3-colorable for {i,j}⊆{3,4,6}.

The $(\Delta,D)$ (degree/diameter) problem consists of finding the largest possible number of vertices $n$ among all the graphs with maximum degree $\Delta$ and diameter $D$. We consider the $(\Delta,D)$ problem for maximal planar bipartite graphs, that is, simple planar graphs in which every face is a quadrangle. We obtain that for the $(\Delta,2)$ problem, the number of vertices is $n=\Delta+2$; and for the $(\Delta,3)$ problem, $n= 3\Delta-1$ if $\Delta$ is odd and $n= 3\Delta-2$ if $\Delta$ is even. Then, we prove that, for the general case of the $(\Delta,D)$ problem, an upper bound on $n$ is approximately $3(2D+1)(\Delta-2)^{\lfloor D/2\rfloor}$, and another one is $C(\Delta-2)^{\lfloor D/2\rfloor}$ if $\Delta\geq D$ and $C$ is a sufficiently large constant. Our upper bounds improve for our kind of graphs the one given by Fellows, Hell and Seyffarth for general planar graphs. We also give a lower bound on $n$ for maximal planar bipartite graphs, which is approximately $(\Delta-2)^{k}$ if $D=2k$, and $3(\Delta-3)^k$ if $D=2k+1$, for $\Delta$ and $D$ sufficiently large in both cases.

The classical theorem of Fary states that every planar graph can be represented by an embedding in which every edge is represented by a straight line segment. We consider generalizations of Fary’s theorem to surfaces equipped with Riemannian metrics. In this setting, we require that every edge is drawn as a shortest path between its two endpoints and we call an embedding with this property a shortest path embedding. The main question addressed in this paper is whether given a closed surface S, there exists a Riemannian metric for which every topologically embeddable graph admits a shortest path embedding. This question is also motivated by various problems regarding crossing numbers on surfaces. We observe that the round metrics on the sphere and the projective plane have this property. We provide flat metrics on the torus and the Klein bottle which also have this property. Then we show that for the unit square flat metric on the Klein bottle there exists a graph without shortest path embeddings. We show, moreover, that for large g, there exist graphs G embeddable into the orientable surface of genus g, such that with large probability a random hyperbolic metric does not admit a shortest path embedding of G, where the probability measure is proportional to the Weil–Petersson volume on moduli space. Finally, we construct a hyperbolic metric on every orientable surface S of genus g, such that every graph embeddable into S can be embedded so that every edge is a concatenation of at most O(g) shortest paths.

Edge Separators of Planar and Outerplanar Graphs With Applications

A graph is ‐colorable if its vertex set can be partitioned into sets , such that for each , the subgraph of induced by has maximum degree at most . The Four Color Theorem states that every planar graph is ‐colorable, and a classical result of Cowen, Cowen, and Woodall shows that every planar graph is ‐colorable. In this paper, we extend both of these results to graphs on surfaces. Namely, we show that every graph embeddable on a surface of Euler genus is ‐colorable and ‐colorable. Moreover, these graphs are also ‐colorable and ‐colorable. We also prove that every triangle‐free graph that is embeddable on a surface of Euler genus is ‐colorable. This is an extension of Grötzsch's Theorem, which states that triangle‐free planar graphs are ‐colorable. Finally, we prove that every graph of girth at least 7 that is embeddable on a surface of Euler genus is ‐colorable. All these results are best possible in several ways as the girth condition is sharp, the constant maximum degrees cannot be improved, and the bounds on the maximum degrees depending on are tight up to a constant multiplicative factor.

Plane graphs with maximum degree 9 are entirely 11-choosable

본 논문은 NP-완전 문제인 간선 색칠과 그래프 부류 결정 문제를 동시에 해결하는 O(E)의 다항시간 알고리즘을 제안하였다. 제안된 알고리즘은 최대차수-최소차수 정점 쌍 간선을 단순히 선택하는 방법으로 간선 채색수 <TEX>${\chi}^{\prime}(G)$</TEX>를 결정하였다. 결정된 <TEX>${\chi}^{\prime}(G)$</TEX>는 <TEX>${\Delta}(G)$</TEX> 또는 <TEX>${\Delta}(G)+1$</TEX>을 얻는다. 결국, 알고리즘 수행 결과 얻은 <TEX>${\chi}^{\prime}(G)$</TEX>로부터 <TEX>${\chi}^{\prime}(G)={\Delta}(G)$</TEX>이면 부류 1, <TEX>${\chi}^{\prime}(G)={\Delta}(G)+1$</TEX>이면 부류 2로 분류할 수 있다. 또한, 미해결 문제로 알려진 "최대차수가 6인 단순, 평면 그래프는 부류 1이다."라는 Vizing의 평면 그래프 추정도 증명하였다. This paper proposes a O(E) polynomial-time algorithm that has been devised to simultaneously solve edge-coloring problem and graph classification problem both of which remain NP-complete. The proposed algorithm selects an edge connecting maximum and minimum degree vertices so as to determine the number of edge coloring <TEX>${\chi}^{\prime}(G)$</TEX>. Determined <TEX>${\chi}^{\prime}(G)$</TEX> is in turn either <TEX>${\Delta}(G)$</TEX> or <TEX>${\Delta}(G)+1$</TEX>. Eventually, the result could be classified as class 1 if <TEX>${\chi}^{\prime}(G)={\Delta}(G)$</TEX> and as category 2 if <TEX>${\chi}^{\pri</TEX><TEX>me}(G)={\Delta}(G)+1$</TEX>. This paper also proves Vizing's planar graph conjecture, which states that 'all simple, planar graphs with maximum degree six or seven are of class one, closing the remaining possible case', which has known to be NP-complete.

An injective coloring of a graph $$G$$ is an assignment of colors to the vertices of $$G$$ so that any two vertices with a common neighbor receive distinct colors. Let $$\chi _{i}^{l}(G)$$ denote the list injective chromatic number of $$G$$ . We prove that (1) $$\chi _{i}^{l}(G)=\Delta $$ for a graph $$G$$ with the maximum average degree $$Mad(G)\le \frac{18}{7}$$ and maximum degree $$\Delta \ge 9$$ ; (2) $$\chi _{i}^{l}(G)\le \Delta +2$$ if $$G$$ is a plane graph with $$\Delta \ge 21$$ and without 3-, 4-, 8-cycles.

Whitney's theorem states that every 3-connected planar graph is uniquely embeddable on the sphere. On the other hand, it has many inequivalent embeddings on another surface. We shall characterize structures of a $3$-connected $3$-regular planar graph $G$ embedded on the projective-plane, the torus and the Klein bottle, and give a one-to-one correspondence between inequivalent embeddings of $G$ on each surface and some subgraphs of the dual of $G$ embedded on the sphere. These results enable us to give explicit bounds for the number of inequivalent embeddings of $G$ on each surface, and propose effective algorithms for enumerating and counting these embeddings.Comment: 19 pages, 12 figures

Improved square coloring of planar graphs

Planar graphs with maximum degree 8 and without adjacent triangles are 9-totally-colorable

We consider dynamic subgraph connectivity problems for planar undirected graphs. In this model there is a fixed underlying planar graph, where each edge and vertex is either off (failed) or (recovered). We wish to answer connectivity queries with respect to the subgraph. The model has two natural variants, one in which there are d edge/vertex failures that precede all connectivity queries, and one in which failures/recoveries and queries are intermixed. We present a d-failure connectivity oracle for planar graphs that processes any d edge/vertex failures in sort(d,n) time so that connectivity queries can be answered in pred(d,n) time. (Here sort and pred are the time for integer sorting and integer predecessor search over a subset of [n] of size d.) Our algorithm has two discrete parts. The first is an algorithm tailored to triconnected planar graphs. It makes use of Barnette's theorem, which states that every triconnected planar graph contains a degree-3 spanning tree. The second part is a generic reduction from general (planar) graphs to triconnected (planar) graphs. Our algorithm is, moreover, provably optimal. An implication of Pǎtrascu and Thorup's lower bound on predecessor search is that no d-failure connectivity oracle (even on trees) can beat pred(d,n) query time. We extend our algorithms to the subgraph connectivity model where edge/vertex failures (but no recoveries) are intermixed with connectivity queries. In triconnected planar graphs each failure and query is handled in O(logn) time (amortized), whereas in general planar graphs both bounds become O(log2n).