Graphs and Combinatorics最新文献

A graph is n-existentially closed if, for all disjoint sets of vertices A and B with (|Acup B|=n), there is a vertex z not in (Acup B) adjacent to each vertex of A and to no vertex of B. In this paper, we investigate n-existentially closed line graphs. In particular, we present necessary conditions for the existence of such graphs as well as constructions for finding infinite families of such graphs. We also prove that there are exactly five 2-existentially closed planar line graphs. We then consider the existential closure of the line graphs of hypergraphs and present constructions for 2-existentially closed line graphs of hypergraphs.

Let t be a positive real number. A graph is called t-tough if the removal of any vertex set S that disconnects the graph leaves at most |S|/t components. The toughness of a graph is the largest t for which the graph is t-tough. We prove that toughness is fixed-parameter tractable parameterized with the treewidth. More precisely, we give an algorithm to compute the toughness of a graph G with running time ({mathcal {O}}(|V(G)|^3cdot textrm{tw}(G)^{2textrm{tw}(G)})) where (textrm{tw}(G)) is the treewidth. If the treewidth is bounded by a constant, then this is a polynomial algorithm.

Let (mathcal {H}) be a set of graphs. The planar Turán number, (ex_mathcal {P}(n,mathcal {H})), is the maximum number of edges in an n-vertex planar graph which does not contain any member of (mathcal {H}) as a subgraph. When (mathcal {H}={H}) has only one element, we usually write (ex_mathcal {P}(n,H)) instead. The study of extremal planar graphs was initiated by Dowden (J Graph Theory 83(3):213–230, 2016). He obtained sharp upper bounds for both (ex_mathcal {P}(n,C_5)) and (ex_mathcal {P}(n,K_4)). Later on, sharp upper bounds were proved for (ex_mathcal {P}(n,C_6)) and (ex_mathcal {P}(n,C_7)). In this paper, we show that (ex_mathcal {P}(n,{K_4,C_5})le {15over 7}(n-2)) and (ex_mathcal {P}(n,{K_4,C_6})le {7over 3}(n-2)). We also give constructions which show the bounds are sharp for infinitely many n.

An equitable k-partition ((kge 2)) of a vertex set S is a partition of S into k subsets (may be empty sets) such that the sizes of any two subsets of S differ by at most one. A local-equitable k-coloring ((kge 2)) of G is an assignment of k colors to the vertices of G such that, for every maximal clique H of G, the coloring on H forms an equitable k-partition of H. Local-equitable coloring of graphs is a generalization of the proper vertex coloring of graphs and also a stronger version of clique-coloring of graphs. Claw-free graphs with maximum degree four are proved to be 2-clique-colorable [Discrete Math. Theoret. Comput. Sci. 11 (2) (2009), 15–24] but not necessary local-equitably 2-colorable. In this paper, given a claw-free graph G with maximum degree at most four, we present a linear time algorithm to give a local-equitable 2-coloring of G or decide that G is not local-equitably 2-colorable. As a corollary, we get that claw-free perfect graphs with maximum degree at most four are local-equitably 2-colorable.

The notion of a k-11-representable graph was introduced by Jeff Remmel in 2017 and studied by Cheon et al. in 2019 as a natural extension of the extensively studied notion of word-representable graphs, which are precisely 0-11-representable graphs. A graph G is k-11-representable if it can be represented by a word w such that for any edge (resp., non-edge) xy in G the subsequence of w formed by x and y contains at most k (resp., at least (k+1)) pairs of consecutive equal letters. A remarkable result of Cheon at al. is that any graph is 2-11-representable, while it is unknown whether every graph is 1-11-representable. Cheon et al. showed that the class of 1-11-representable graphs is strictly larger than that of word-representable graphs, and they introduced a useful toolbox to study 1-11-representable graphs. In this paper, we introduce new tools for studying 1-11-representation of graphs. We apply them for establishing 1-11-representation of Chvátal graph, Mycielski graph, split graphs, and graphs whose vertices can be partitioned into a comparability graph and an independent set.

The vertex arboricity a(G) of a graph G is the minimum number of colors required to color the vertices of G such that no cycle is monochromatic. A graph G is 1-planar if it can be drawn in the plane so that each edge has at most one crossing. In this paper, we proved that every 1-planar graph without 5-cycles has minimum degree at most 5; Every 1-planar graph of girth at least 7 has minimum degree at most 3. The following conclusions can be obtained by combining the existing conclusions and our proofs: if G is a 1-planar graph without 5-cycles, then (a(G)le 3); if G is a 1-planar graph with (g(G)ge 7), then (a(G)le 2).

We consider two different notions of graph colouring, namely, the t-periodic colouring for vertices that has been introduced in 1974 by Bondy and Simonovits, and the periodic colouring for oriented edges that has been recently introduced in the context of spectral theory of non-backtracking operators. For each of these two colourings, we introduce the corresponding colouring number which is given by maximising the possible number of colours. We first investigate these two new colouring numbers individually, and we then show that there is a deep relationship between them.

Let ({mathcal {C}}_{k}(n)) denote the family of all connected graphs of order n with chromatic number k. In this paper we show that the conjecture proposed by Tomescu which if (xge kge 4) and (Gin {mathcal {C}}_{k}(n)), then

holds under the additional condition that G has an independent cut-set T of size at most 2 such that the number of components in (G{setminus } T) is equal to the independence number of G.

For a graph G, let (R({mathcal {C}}(nG))) denote the least N such that every 2-colouring of the edges of (K_N) contains a monochromatic copy of nG in a monochromatic connected subgraph, where nG denotes n vertex disjoint copies of G. Gyárfás and Sárközy (J Graph Theory 83(2):109–119, 2016) showed that (R({mathcal {C}}(nK_3))=7n-2) for (n ge 2). After that, Roberts (Electron J Comb 24(1):8, 2017)showed that (R({mathcal {C}}(nK_r))=(r^2-r+1)n-r+1) for (r ge 4) and (n ge R(K_r)), where (R(K_r)) is the Ramsey number of (K_r). In this paper, we determine (R({mathcal {C}}(nG))) for all 4-vertex graphs G without isolated vertices.

In this paper, we give tight bounds for the normalized Laplacian eigenvalues of hypergraphs that are not necessarily uniform, and provide an edge version interlacing theorem, a Cheeger inequality, and a discrepancy inequality that are related to the normalized Laplacian eigenvalues for uniform hypergraphs.