Labeling planar graphs with a condition at distance two

An L(2, 1)-labelling of a graph is a function f from the vertex set to the positive integers such that |f(x) - f(y)| ≥ 2 if dist(x, y) = 1 and |f(x) - f(y)| ≥ 1 if dist(x, y) = 2, where dist(x, y) is the distance between the two vertices x and y in the graph G. The span of an L(2, 1)-labelling f is the difference between the largest and the smallest labels used by f plus 1. In 1992, Griggs and Yeh conjectured that every graph with maximum degree Δ ≥ 2 has an L(2, 1)-labelling with span at most Δ2 + 1. By settling this conjecture for Δ sufficiently large, we prove the existence of a constant C such that the span of any graph of maximum degree Δ is at most Δ 2 + C.

Equitable colorings of planar graphs with maximum degree at least nine

Recently Sun et al [X.-Y. Sun, J.-L. Wu, Y.-W. Wu, J.-F. Hou, Total colorings of planar graphs without adjacent triangles, Discrete Math 309:202-206 (2009)] proved that planar graphs with maximum degree six and with no adjacent triangles are total $8$-colorable. This results implies that if every vertex of a planar graph of maximum degree six is missing either a $3$-cycle or a $4$-cycle, then the graph is total $8$-colorable. In this paper we strengthen that condition by showing that if every vertex of a planar graph of maximum degree six is missing some $k_v$-cycle for $k_v \\in \\{3,4,5,6,7,8\\}$, then the graph is total $8$-colorable.

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.

Edge Separators of Planar and Outerplanar Graphs With Applications

The computational complexity of cordial and equitable labelling

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.

Plane graphs with maximum degree 9 are entirely 11-choosable

One important problem in graph theory is graph coloring or graph labeling.Labeling problem is a well-studied problem due to its wide applications, especially in frequency assignment in(mobile) communication systems, coding theory, x-ray crystallography, radar, circuit design, etc.For two non-negative integers, labeling of a graph is a function from the node set to the set of non-negative integers.For any two distinct vertices and of , a radio labeling is an injective function : 0 such that , | | 1 where is the diameter of .The radio number of , is the maximum value of taken overall radio labelings of .This paper determines the radio number of double triangular snake graphs.

This paper initiates the study of the maximum neighborhood degree realization problem. Given a sequence D=(d1,…,dn) of non-negative integers, the goal is to construct a simple graph with vertices v1,…,vn such that for every i∈[1,n], the maximum degree in the neighborhood of vi is exactly di (or output null if no such graph exists). Depending upon whether or not the realizing graph is required to be connected, and whether or not the neighborhood of a vertex is closed (that is, the neighborhood includes the vertex itself), the problem has four natural settings. We provide complete realizability criteria for all four settings of the problem. Our conditions are verifiable in linear time and our realizations can be constructed in polynomial time. In addition, we prove tight/approximate bounds for the number of maximum neighboring degree profiles of length n that are realizable.

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.

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.