Graphs and Combinatorics最新文献

Abstract

The planar Turán number of a graph H, denoted by (ex_{_mathcal {P}}(n,H)) , is the maximum number of edges in a planar graph on n vertices without containing H as a subgraph. This notion was introduced by Dowden in 2016 and has attracted quite some attention since then; those work mainly focus on finding (ex_{_mathcal {P}}(n,H)) when H is a cycle or Theta graph or H has maximum degree at least four. In this paper, we first completely determine the exact values of (ex_{_mathcal {P}}(n,H)) when H is a cubic graph. We then prove that (ex_{_mathcal {P}}(n,2C_3)=lceil 5n/2rceil -5) for all (nge 6) , and obtain the lower bounds of (ex_{_mathcal {P}}(n,2C_k)) for all (nge 2kge 8) . Finally, we also completely determine the exact values of (ex_{_mathcal {P}}(n,K_{2,t})) for all (tge 3) and (nge t+2) .

A graph is k-vertex-critical if (chi (G)=k) but (chi (G-v)<k) for all (vin V(G)). We construct new infinite families of k-vertex-critical ((P_5,C_5))-free graphs for all (kge 6). Our construction generalises known constructions for 4-vertex-critical (P_7)-free graphs and 5-vertex-critical (P_5)-free graphs and is in contrast to the fact that there are only finitely many 5-vertex-critical ((P_5,C_5))-free graphs. In fact, our construction is even more well-structured, being ((2P_2,K_3+P_1,C_5))-free.

According to Ramsey’s Theorem, for any natural p and q there is a minimum number R(p, q) such that every graph with at least R(p, q) vertices has either a clique of size p or an independent set of size q. In the present paper, we study Ramsey numbers R(p, q) for graphs in special classes. It is known that for graphs of bounded co-chromatic number Ramsey numbers are upper-bounded by a linear function of p and q. However, the exact values of R(p, q) are known only for classes of graphs of co-chromatic number at most 2. In this paper, we determine the exact values of Ramsey numbers for classes of graphs of co-chromatic number at most 3. It is also known that for classes of graphs of unbounded splitness the value of R(p, q) is lower-bounded by ((p-1)(q-1)+1). This lower bound coincides with the upper bound for perfect graphs and for all their subclasses of unbounded splitness. We call a class Ramsey-perfect if there is a constant c such that (R(p,q)=(p-1)(q-1)+1) for all (p,qge c) in this class. In the present paper, we identify a number of Ramsey-perfect classes which are not subclasses of perfect graphs.

The k-weak-dynamic number of a graph G is the smallest number of colors we need to color the vertices of G in such a way that each vertex v of degree d(v) sees at least min({k,d(v)}) colors on its neighborhood. We use reducible configurations and list coloring of graphs to prove that all planar graphs have 3-weak-dynamic number at most 6.

Given an edge-colored graph (G_c), a set of p pairs of vertices ((a_i,b_i)) together with p numbers (k_1,k_2, ldots k_p) associated with the pairs, can we find a set of alternating paths linking the pairs ((a_1,b_1)), ((a_2,b_2), ldots ), in their respective numbers (k_1,k_2,ldots k_p)? Such is the question addressed in this paper. The problem being highly intractable, we consider a restricted version of it to edge-colored complete graphs. Even so restricted, the problem remains intractable if the paths/trails must be edge-disjoint, but it ceases to be so if the paths/trails are to be vertex-disjoint, as is proved in this paper. An approximation algorithm is presented in the end, with a performance ratio asymptotically close to 3/4 for a restricted version of the problem.

The Borodin–Kostochka Conjecture states that for a graph G, if (Delta (G)ge 9), then (chi (G)le max {Delta (G)-1,omega (G)}). In this paper, we prove the Borodin–Kostochka Conjecture holding for odd-hole-free graphs.

We deal with a variant of the 1–2–3 Conjecture introduced by Gao, Wang, and Wu (Graphs Combin 32:1415–1421, 2016) . This variant asks whether all graphs can have their edges labelled with 1 and 2 so that when computing the sums of labels incident to the vertices, no monochromatic cycle appears. In the aforementioned seminal work, the authors mainly verified their conjecture for a few classes of graphs, namely graphs with maximum average degree at most 3 and series–parallel graphs, and observed that it also holds for simple classes of graphs (cycles, complete graphs, and complete bipartite graphs). In this work, we provide a deeper study of this conjecture, establishing strong connections with other, more or less distant notions of graph theory. While this conjecture connects quite naturally to other notions and problems surrounding the 1–2–3 Conjecture, it can also be expressed so that it relates to notions such as the vertex-arboricity of graphs. Exploiting such connections, we provide easy proofs that the conjecture holds for bipartite graphs and 2-degenerate graphs, thus generalising some of the results of Gao, Wang, and Wu. We also prove that the conjecture holds for graphs with maximum average degree less than (frac{10}{3}), thereby strengthening another of their results. Notably, this also implies the conjecture holds for planar graphs with girth at least 5. All along the way, we also raise observations and results highlighting why the conjecture might be of greater interest.

For graphs F, G, and H, we write (F rightarrow (G,H)) if every red-blue coloring of the edges of F produces a red copy of G or a blue copy of H. The graph F is said to be (G, H)-minimal if it is subgraph-minimal with respect to this property. The characterization problem for Ramsey-minimal graphs is classically done for finite graphs. In 2021, Barrett and the second author generalized this problem to infinite graphs. They asked which pairs (G, H) admit a Ramsey-minimal graph and which ones do not. We show that any pair of star forests such that at least one of them involves an infinite-star component admits no Ramsey-minimal graph. Also, we construct a Ramsey-minimal graph for a finite star forest versus a subdivision graph. This paper builds upon the results of Burr et al. (Discrete Math 33:227–237, 1981) on Ramsey-minimal graphs for finite star forests.

We consider multipermutations and a certain partial order, the weak Bruhat order, on this set. This generalizes the Bruhat order for permutations, and is defined in terms of containment of inversions. Different characterizations of this order are given. We also study special multipermutations called Stirling multipermutations and their properties.

Abstract

Lattice path matroids and bicircular matroids are two well-known classes of transversal matroids. In the seminal work of Bonin and de Mier about structural properties of lattice path matroids, the authors claimed that lattice path matroids significantly differ from bicircular matroids. Recently, it was proved that all cosimple lattice path matroids have positive double circuits, while it was shown that there is a large class of cosimple bicircular matroids with no positive double circuits. These observations support Bonin and de Miers’ claim. Finally, Sivaraman and Slilaty suggested studying the intersection of lattice path matroids and bicircular matroids as a possibly interesting research topic. In this work, we exhibit the excluded bicircular matroids for the class of lattice path matroids, and we propose a characterization of the graph family whose bicircular matroids are lattice path matroids. As an application of this characterization, we propose a geometric description of 2-connected lattice path bicircular matroids.