Graphs and Combinatorics最新文献

Let ([n]=X_1cup X_2cup X_3) be a partition with (lfloor frac{n}{3}rfloor le |X_i|le lceil frac{n}{3}rceil ) and define ({mathcal {G}}={Gsubset [n]:|Gcap X_i|le 1, 1le ile 3}). It is easy to check that the trace ({mathcal {G}}_{mid Y}:={Gcap Y:Gin {mathcal {G}}}) satisfies (|{mathcal {G}}_{mid Y}|le 12) for all 4-sets (Ysubset [n]). In the present paper, we prove that if ({mathcal {F}}subset 2^{[n]}) satisfies (|{mathcal {F}}|>|{mathcal {G}}|) and (nge 28), then (|{mathcal {F}}_{mid C}|ge 13) for some (Csubset [n]), (|C|=4). Several further results of a similar flavor are established as well.

A weak-dynamic coloring of a graph is a vertex coloring (not necessarily proper) in such a way that each vertex of degree at least two sees at least two colors in its neighborhood. It is proved that the weak-dynamic chromatic number of the class of k-planar graphs (resp. IC-planar graphs) is equal to (resp. at most) the chromatic number of the class of 2k-planar graphs (resp. 1-planar graphs), and therefore every IC-planar graph has a weak-dynamic 6-coloring (being sharp) and every 1-planar graph has a weak-dynamic 9-coloring. Moreover, we conclude that the well-known Four Color Theorem is equivalent to the proposition that every planar graph has a weak-dynamic 4-coloring, or even that every (C_4)-free bipartite planar graph has a weak-dynamic 4-coloring. It is also showed that deciding if a given graph has a weak-dynamic k-coloring is NP-complete for every integer (kge 3).

Abstract

Let (P_4) denote the path on four vertices. A (P_4) -packing of a graph G is a collection of vertex-disjoint copies of (P_4) in G. The maximum (P_4) -packing problem is to find a (P_4) -packing of maximum cardinality in a graph. In this paper, we prove that every simple cubic graph G on v(G) vertices has a (P_4) -packing covering at least (frac{2v(G)}{3}) vertices of G and that this lower bound is sharp. Our proof provides a quadratic-time algorithm for finding such a (P_4) -packing of a simple cubic graph.

For graphs (H_1,H_2,dots ,H_k), the k-color Turán number (ex(n,H_1,H_2,dots ,H_k)) is the maximum number of edges in a k-colored graph of order n that does not contain monochromatic (H_i) in color i as a subgraph, where (1le ile k). In this note, we show that if (H_i) is a bipartite graph with at least two edges for (1le ile k), then (ex(n,H_1,H_2,dots ,H_k)=(1+o(1))sum _{i=1}^kex(n,H_i)) as (nrightarrow infty ), in which the non-constructive proof for some cases can be derandomized.

Tashkinov-trees have been used as a tool for proving bounds on the chromatic index, and are becoming a fundamental tool for edge-coloring. Was its publication in a language different from English an obstacle for the accessibility of a clean and complete proof of Tashkinov’s fundamental theorem? Tashkinov’s original, Russian paper offers a clear presentation of this theorem and its proof. The theorem itself has been well understood and successfully applied, but the proof is more difficult. It builds a truly amazing recursive machine, where the various cases necessitate a refined and polished analysis to fit into one another with surprising smoothness and accuracy. The difficulties were brilliantly unknotted by the author, deserving repeated attention. The present work is the result of reading, translating, reorganizing, rewriting, completing, shortcutting and annotating Tashkinov’s proof. It is essentially the same proof, with non-negligeable communicational differences though, for instance completing it wherever it occurred to be necessary, and simplifying it whenever it appeared to be possible, at the same time trying to adapt it to the habits and taste of the international graph theory community.

Given a graph H and a positive integer n, the Turán number of H of the order n, denoted by ex(n, H), is the maximum size of a simple graph of order n that does not contain H as a subgraph. Given graphs G and H, (G vee H) denotes the join of G and H. In this paper, we prove (ex(n, K_m vee C_{2k-1}) = leftlfloor frac{(m+1)n^2}{2(m+2)}rightrfloor ) for (nge 2(m+2)k-3(m+2)-1).

A graph that can be generated from (K_1) using joins and 0-sums is called a cograph. We define a sesquicograph to be a graph that can be generated from (K_1) using joins, 0-sums, and 1-sums. We show that, like cographs, sesquicographs are closed under induced minors. Cographs are precisely the graphs that do not have the 4-vertex path as an induced subgraph. We obtain an analogue of this result for sesquicographs, that is, we find those non-sesquicographs for which every proper induced subgraph is a sesquicograph.

A connected graph G is called (l_{1})-embeddable, if it can be isometrically embedded into the (l_{1})-space. The shifted quadrilateral cylinder graph (O_{m,n,k}) is a class of quadrilateral tilings on a cylinder obtained by rolling the grid graph (P_{m}square P_{n}) via shifting k positions. In this article, we determine that all the (O_{m,n,k}) are not (l_{1})-embeddable except for (O_{m,n,0}) and (O_{m,3,1}).

Let (G=(V,E)) be a graph with no isolated vertices. A vertex v totally dominates a vertex w ((w ne v)), if v is adjacent to w. A set (D subseteq V) called a total dominating set of G if every vertex (vin V) is totally dominated by some vertex in D. The minimum cardinality of a total dominating set is the total domination number of G and is denoted by (gamma _t(G)). A total dominator coloring of graph G is a proper coloring of vertices of G, so that each vertex totally dominates some color class. The total dominator chromatic number (chi _{{textrm{td}}}(G)) of G is the least number of colors required for a total dominator coloring of G. The Total Dominator Coloring problem is to find a total dominator coloring of G using the minimum number of colors. It is known that the decision version of this problem is NP-complete for general graphs. We show that it remains NP-complete even when restricted to bipartite, planar and split graphs. We further study the Total Dominator Coloring problem for various graph classes, including trees, cographs and chain graphs. First, we characterize the trees having (chi _{{textrm{td}}}(T)=gamma _t(T)+1), which completes the characterization of trees achieving all possible values of (chi _{{textrm{td}}}(T)). Also, we show that for a cograph G, (chi _{{textrm{td}}}(G)) can be computed in linear-time. Moreover, we show that (2 le chi _{{textrm{td}}}(G) le 4) for a chain graph G and then we characterize the class of chain graphs for every possible value of (chi _{{textrm{td}}}(G)) in linear-time.

DP-coloring was introduced by Dvořák and Postle as a generalization of list coloring and signed coloring. A new coloring, strictly f-degenerate transversal, is a further generalization of DP-coloring and L-forested-coloring. In this paper, we present some structural results on planar and toroidal graphs with forbidden configurations, and establish some sufficient conditions for the existence of strictly f-degenerate transversal based on these structural results. Consequently, (i) every toroidal graph without subgraphs in Fig. 2 is DP-4-colorable, and has list vertex arboricity at most 2, (ii) every toroidal graph without 4-cycles is DP-4-colorable, and has list vertex arboricity at most 2, (iii) every planar graph without subgraphs isomorphic to the configurations in Fig. 3 is DP-4-colorable, and has list vertex arboricity at most 2. These results improve upon previous results on DP-4-coloring (Kim and Ozeki in Discrete Math 341(7):1983–1986. https://doi.org/10.1016/j.disc.2018.03.027, 2018; Sittitrai and Nakprasit in Bull Malays Math Sci Soc 43(3):2271–2285. https://doi.org/10.1007/s40840-019-00800-1, 2020) and (list) vertex arboricity (Choi and Zhang in Discrete Math 333:101–105. https://doi.org/10.1016/j.disc.2014.06.011, 2014; Huang et al. in Int J Math Stat 16(1):97–105, 2015; Zhang in Iranian Math Soc 42(5):1293–1303, 2016).