A note on a triangle-free — complete graph induced Ramsey number

The random greedy algorithm for constructing a large partial Steiner-Triple-System is defined as follows. Begin with a complete graph on n vertices and proceed to remove the edges of triangles one at a time, where each triangle removed is chosen uniformly at random out of all remaining triangles. This stochastic process terminates once it arrives at a triangle-free graph, and a longstanding open problem is to estimate the final number of edges, or equivalently the time it takes the process to conclude. The intuition that the edge distribution is roughly uniform at all times led to a folklore conjecture that the final number of edges is n 3/2+o(1) with high probability, whereas the best known upper bound is n 7/4+o(1) . It is no coincidence that various methods break precisely at the exponent 7/4 as it corresponds to the inherent barrier where co-degrees become comparable to the variations in their values that arose earlier in the process. In this work we significantly improve upon the previous bounds by establishing that w.h.p. the number of edges in the final graph is at most n 5/3+o(1) . Our approach relies on a system of martingales used to control key graph parameters, where the crucial new idea is to harness the self-correcting nature of the process in order to control these parameters well beyond the point where their early variation matches the order of their expectation.

Random triangle removal

Tuza’s Conjecture asserts that the minimum number $$\tau _{\varDelta }'(G)$$ of edges of a graph G whose deletion results in a triangle-free graph is at most 2 times the maximum number $$\nu _{\varDelta }'(G)$$ of edge-disjoint triangles of G. The complete graphs $$K_{4}$$ and $$K_{5}$$ show that the constant 2 would be best possible. Moreover, if true, the conjecture would be essentially tight even for $$K_{4}$$ -free graphs. In this paper, we consider several subclasses of $$K_{4}$$ -free graphs. We show that the constant 2 can be improved for them and we try to provide the optimal one. The classes we consider are of two kinds: graphs with edges in few triangles and graphs obtained by forbidding certain odd-wheels. We translate an approximate min-max relation for $$\tau _{\varDelta }'(G)$$ and $$\nu _{\varDelta }'(G)$$ into an equivalent one for the clique cover number and the independence number of the triangle graph of G and we provide $$\theta $$ -bounding functions for classes related to triangle graphs. In particular, we obtain optimal $$\theta $$ -bounding functions for the classes $${\textit{Free}}(K_{5}, \text{ claw, } \text{ diamond })$$ and $${\textit{Free}}(P_{5}, \text{ diamond }, K_{2,3})$$ and a $$\chi $$ -bounding function for the class $$(\text{ banner, } \text{ odd-hole }, \overline{K_{1, 4}})$$ .

It is well-known that eigenvalues of graphs can be used to describe structural properties and parameters of graphs. A theorem of Nosal and Nikiforov states that if $G$ is a triangle-free graph with $m$ edges, then $\lambda (G)\le \sqrt{m}$, equality holds if and only if $G$ is a complete bipartite graph. Recently, Lin, Ning and Wu [Combin. Probab. Comput. 30 (2021)] proved a generalization for non-bipartite triangle-free graphs. Moreover, Zhai and Shu [Discrete Math. 345 (2022)] presented a further improvement. In this paper, we present an alternative method for proving the improvement by Zhai and Shu. Furthermore, the method can allow us to give a refinement on the result of Zhai and Shu for non-bipartite graphs without short odd cycles.

An induced matching in a graph is a set of edges whose endpoints induce a \(1\)-regular subgraph. It is known that every \(n\)-vertex graph has at most \(10^{n/5}\approx 1.5849^n\) maximal induced matchings, and this bound is best possible. We prove that every \(n\)-vertex triangle-free graph has at most \(3^{n/3}\approx 1.4423^n\) maximal induced matchings, and this bound is attained by every disjoint union of copies of the complete bipartite graph \(K_{3,3}\). Our result implies that all maximal induced matchings in an \(n\)-vertex triangle-free graph can be listed in time \(O(1.4423^n)\), yielding the fastest known algorithm for finding a maximum induced matching in a triangle-free graph.KeywordsMaximum Induced MatchingTriangle-free GraphsMaximum Independent SetTwin SetsPolynomial DelayThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We present upper bounds on the diameter of bipartite and triangle-free graphs with prescribed edge connectivity with respect to order and size. All bounds presented in this paper are asymptotically sharp.

SIZE RAMSEY NUMBERS INVOLVING MATCHINGS

A finite and simple graph is said to be equitably colorable if its vertices can be partitioned into classes such that each is an independent set and holds for every . The smallest integer for which is equitable chromatic number of and denoted by . The equitable chromatic threshold of a graph , denoted by , is the minimum such that is equitably colorable for all . This paper focuses on the equitable colorability of rooted product of graphs, in particular, exact values or upper bounds of and when and are cycles, paths, complete graphs and complete partite graphs have been found.

The mixed irredundant Ramsey number t(m, n) is the smallest natural number t such that if the edges of the complete graph Kt on t vertices are arbitrarily bi-coloured using the colours blue and red, then necessarily either the subgraph induced by the blue edges has an irredundant set of cardinality m or the subgraph induced by the red edges has an independent set of cardinality n (or both). Previously it was known that 18 ≤ t(3, 7) ≤ 22 and 18 ≤ t(3, 8) ≤ 28. In this paper we prove that t(3, 7) = 18 and t(3, 8) = 22.

A star coloring of a graph G is a proper vertex coloring in which every path on four vertices in G is not bicolored. The star chromatic number χs (G) of G is the least number of colors needed to star color G. Let G = (V,E) be a graph with V = S1 [ S2 [ S3 [ . . . [ St [ T where each Si is a set of all vertices of the same degree with at least two elements and T =V (G) − St i=1 Si. The degree splitting graph DS (G) is obtained by adding vertices w1,w2, . . .wt and joining wi to each vertex of Si for 1 i t. The comb product between two graphs G and H, denoted by G ⊲ H, is a graph obtained by taking one copy of G and |V (G)| copies of H and grafting the ith copy of H at the vertex o to the ith vertex of G. In this paper, we give the exact value of star chromatic number of degree splitting of comb product of complete graph with complete graph, complete graph with path, complete graph with cycle, complete graph with star graph, cycle with complete graph, path with complete graph and cycle with path graph.

The H-free process, for some fixed graph H, is the random graph process defined by starting with an empty graph on n vertices and then adding edges one at a time, chosen uniformly at random subject to the constraint that no H subgraph is formed. Let G be the random maximal H-free graph obtained at the end of the process. When H is strictly 2-balanced, we show that for some c>0, with high probability as n→∞, the minimum degree in G is at least \(cn^{1-(v_{H}-2)/(e_{H}-1)}(\log n)^{1/(e_{H}-1)}\). This gives new lower bounds for the Turán numbers of certain bipartite graphs, such as the complete bipartite graphs K r,r with r≥5. When H is a complete graph K s with s≥5 we show that for some C>0, with high probability the independence number of G is at most \(Cn^{2/(s+1)}(\log n)^{1-1/(e_{H}-1)}\). This gives new lower bounds for Ramsey numbers R(s,t) for fixed s≥5 and t large. We also obtain new bounds for the independence number of G for other graphs H, including the case when H is a cycle. Our proofs use the differential equations method for random graph processes to analyse the evolution of the process, and give further information about the structure of the graphs obtained, including asymptotic formulae for a broad class of subgraph extension variables.

This paper studies algorithmic aspects of majority domination, which is a variation of domination in graph theory. A majority dominating function of a graph $G=(V,E)$ is a function $g$ from $V$ to $\{-1,1\}$ such that $\sum_{u\in N[v]}\,g(u)\geq 1$ for at least half of the vertices $v\in V$. The majority domination problem is to find a majority dominating function $g$ of a given graph $G=(V,E)$ such that $\sum_{v\in V}\,g(v)$ is minimized. The concept of majority domination was introduced by Hedetniemi and studied by Broere {\em et al}., who gave exact values for the majority domination numbers of complete graphs, complete bipartite graphs, paths, and unions of two complete graphs. They also proved that the majority domination problem is NP-complete for general graphs; and asked if the problem NP-complete for trees. The main result of this paper is to give polynomial-time algorithms for the majority domination problem in trees, cographs, and $k$-trees with fixed $k$.

Using computational techniques we derive six new upper bounds on the classical two-color Ramsey numbers: $R(3,10) \le 42$, $R(3,11) \le 50$, $R(3,13) \le 68$, $R(3,14) \le 77$, $R(3,15) \le 87$, and $R(3,16) \le 98$. All of them are improvements by one over the previously best known bounds. Let $e(3,k,n)$ denote the minimum number of edges in any triangle-free graph on $n$ vertices without independent sets of order $k$. The new upper bounds on $R(3,k)$ are obtained by completing the computation of the exact values of $e(3,k,n)$ for all $n$ with $k \leq 9$ and for all $n \leq 33$ for $k = 10$, and by establishing new lower bounds on $e(3,k,n)$ for most of the open cases for $10 \le k \le 15$. The enumeration of all graphs witnessing the values of $e(3,k,n)$ is completed for all cases with $k \le 9$. We prove that the known critical graph for $R(3,9)$ on 35 vertices is unique up to isomorphism. For the case of $R(3,10)$, first we establish that $R(3,10)=43$ if and only if $e(3,10,42)=189$, or equivalently, that if $R(3,10)=43$ then every critical graph is regular of degree 9. Then, using computations, we disprove the existence of the latter, and thus show that $R(3,10) \le 42$.

Grotzsch's theorem states that every triangle-free planar graph is 3-colorable, and several relatively simple proofs of this fact were provided by Thomassen and other authors. It is easy to convert these proofs into quadratic-time algorithms to find a 3-coloring, but it is not clear how to find such a coloring in linear time (Kowalik used a nontrivial data structure to construct an O(n log n) algorithm).We design a linear-time algorithm to find a 3-coloring of a given triangle-free planar graph. The algorithm avoids using any complex data structures, which makes it easy to implement. As a by-product we give a yet simpler proof of Grotzsch's theorem.

A star coloring of a graph $G$ is a proper vertex coloring in which every path on four vertices in $G$ is not bicolored. The star chromatic number $\chi_{s}\left(G\right)$ of $G$ is the least number of colors needed to star color $G$. In this paper, we find the exact values of the star chromatic number of modular product of complete graph with complete graph $K_m \diamond K_n$, path with complete graph $P_m \diamond K_n$ and star graph with complete graph $K_{1,m}\diamond K_n$. \par All graphs in this paper are finite, simple, connected and undirected graph and we follow \cite{bm, cla, f} for terminology and notation that are not defined here. We denote the vertex set and the edge set of $G$ by $V(G)$ and $E(G)$, respectively. Branko Gr\"{u}nbaum introduced the concept of star chromatic number in 1973. A star coloring \cite{alberton, fertin, bg} of a graph $G$ is a proper vertex coloring in which every path on four vertices uses at least three distinct colors. The star chromatic number $\chi_{s}\left(G\right)$ of $G$ is the least number of colors needed to star color $G$. \par During the years star coloring of graphs has been studied extensively by several authors, for instance see \cite{alberton, col, fertin}.