Covering projective planar graphs with three forests

Equitable colorings of planar graphs with maximum degree at least nine

Links in projective planar graphs

An edge or face of an embedded graph is light if the sum of the degrees of the vertices incident with it is small. This paper parallelizes four inequalities on the number of light edges and light triangles from the plane to the projective plane. Each of the four inequalities is shown to be the best possible. © 1996 John Wiley & Sons, Inc.

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.

Edge Separators of Planar and Outerplanar Graphs With Applications

Plane graphs with maximum degree 9 are entirely 11-choosable

Improved square coloring of planar graphs

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.

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

This paper is devoted to study the combinatorial properties of Local Minimum Spanning Trees (LMSTs), a geometric structure that is attracting increasing research interest in the wireless sensor networks community. Namely, we study which topologies are allowed for a sensor network that uses, for supporting connectivity, a local minimum spanning tree approach. First, we refine the current definition of LMST realizability, focusing on the role of the power of transmission (i.e., of the radius of the covered area). Second, we show simple planar, connected, and triangle-free graphs with maximum degree 3 that cannot be represented as an LMST. Third, we present several families of graphs that can be represented as LMSTs. Then, we show a relationship between planar graphs and their representability as LMSTs based on homeomorphism. Finally, we show that the general problem of determining whether a graph is LMST representable is NP-hard.

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.

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

Although Min-CDS in general graphs is hard to approximate, the restriction to certain special graph classes admits much better approximation results. Min-CDS in planar graphs remains NP-hard even for planar graphs that are regular of degree 4 [57]. The related problem, Min-DS in planar graphs, is also NP-hard even for planar graphs with maximum vertex degree 3 and planar graphs that are regular of degree 4 [57]. It is well known that Min-DS in planar graphs possesses a polynomial-time approximation scheme (PTAS) based on the shifting strategy [3]: For any constant e >0, there is a polynomial-time 1+e-approximation algorithm. Thus, it is immediate to conclude that Min-CDS in planar graphs can be approximated within a factor 3+e for any e>0 in polynomial time. However, the degree of the polynomial grows with 1∕e and hence, the approximation scheme is hardly practical.

We introduce a new method to optimize the required area, minimum angle, and number of bends of planar graph drawings on a grid. The main tool is a new type of ordering on the vertices and faces of triconnected planar graphs. Using this method linear-time-and-space algorithms can be designed for many graph-drawing problems. Our main results are as follows: Every triconnected planar graphG admits a planar convex grid drawing with straight lines on a (2n−4)×(n−2) grid, wheren is the number of vertices. Every triconnected planar graph with maximum degree 4 admits a planar orthogonal grid drawing on ann×n grid with at most [3n/2]+4 bends, and ifn>6, then every edge has at most two bends. Every planar graph with maximum degree 3 admits a planar orthogonal grid drawing with at most [n/2]+1 bends on an [n/2]×[n/2] grid. Every triconnected planar graphG admits a planar polyline grid drawing on a (2n−6)×(3n−9) grid with minimum angle larger than 2/d radians and at most 5n−15 bends, withd the maximum degree. These results give in some cases considerable improvements over previous results, and give new bounds in other cases. Several other results, e.g., concerning visibility representations, are included.