Reverse super magic strength of some special graphs

A reverse magic labelling of a graph G(V, E) is a bijection f: V ∪ E → {1, 2, 3, ......, v + ε} such that for all edges xy, f(xy) - {f(x) + f(y)} is a constant which is denoted by c(f). A reverse magic labelling of a graph G(V, E) is called reverse super edge-magic labelling of G if f(V) = {1, 2, ...... v} and f(E) = {v + 1, v + 2, ......, v + ε}. The reverse super edge-magic strength of a graph G,rsm(G), is defined as the minimum of all c(f) where the minimum is taken over all reverse edge-magic labelling f of G. In this paper we invented the reverse super edge-magic strength of banana trees.

In this paper we disprove a conjecture of Lidický and Murphy about the number of copies of a given graph in a $K_r$-free graph and give an alternative general conjecture. We also prove an asymptotically tight bound on the number of copies of any bipartite graph of radius at most~$2$ in a triangle-free graph.

Let G = ( V , E ) be a finite undirected graph with vertex set V ( G ) of order | V ( G )| = n and edge set E ( G ) of size | E ( G )| = m . Let Δ = d 1 ≥ d 2 ≥ ⋯ ≥ d n = δ be the degree sequence of the graph G . A clique in a graph G is a complete subgraph of G . The clique number of a graph G , denoted by ω ( G ) , is the order of a maximum clique of G . In 1907 Mantel proved that a triangle-free graph with n vertices can contain at most ⌊ n 2 /4⌋ edges. In 1941 Tur'an generalized Mantel’s result to graphs not containing cliques of size r by proving that graphs of order n that contain no induced K r have at most (1 − 1/ r − 1) n 2 /2 edges. In this paper, we give new bounds for the maximum number of edges in a K r -free graph G of order n , minimum degree δ , and maximum degree Δ . We show that, for the families of graphs having the above properties, our bounds are slightly better than the more general bounds of Tur'an.

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 $$(\text{ banner, } \text{ odd-hole }, \overline{K_{1, 4}})$$ .

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.

The VERTEX COLOURING problem is known to be NP-complete in the class of triangle-free graphs. Moreover, it remains NP-complete even if we additionally exclude a graph F which is not a forest. We study the computational complexity of the problem in (K3, F)-free graphs with F being a forest. From known results it follows that for any forest F on 5 vertices the VERTEX COLOURING problem is polynomial-time solvable in the class of (K3, F)-free graphs. In the present paper, we show that the problem is also polynomial-time solvable in many classes of (K3, F)-free graphs with F being a forest on 6 vertices.

The independence number in graphs of maximum degree three

The NP-completeness of chromatic index in triangle free graphs with maximum vertex of degree 3

We show that several problems that are hard for various parameterized complexity classes on general graphs, become fixed parameter tractable on graphs with no small cycles. More specifically, we give fixed parameter tractable algorithms for Dominating Set, t -Vertex Cover (where we need to cover at least t edges) and several of their variants on graphs with girth at least five. These problems are known to be W[i]-hard for some i≥1 in general graphs. We also show that the Dominating Set problem is W[2]-hard for bipartite graphs and hence for triangle free graphs. In the case of Independent Set and several of its variants, we show these problems to be fixed parameter tractable even in triangle free graphs. In contrast, we show that the Dense Subgraph problem where one is interested in finding an induced subgraph on k vertices having at least l edges, parameterized by k, is W[1]-hard even on graphs with girth at least six. Finally, we give an O(log p) ratio approximation algorithm for the Dominating Set problem for graphs with girth at least 5, where p is the size of an optimum dominating set of the graph. This improves the previous O(log n) factor approximation algorithm for the problem, where n is the number of vertices of the input graph.

The number of maximal independent sets in connected triangle-free graphs

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

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

Let $G=(V, E)$ be a graph and let $(A,+)$ be an Abelian group with identity elemento. Let $f: V \rightarrow A$ be a vertex labeling and $f^*: E \rightarrow A$ be the induced labeling of $f$, defined by $f^*\left(v_1 v_2\right)=f\left(v_1\right)+f\left(v_2\right)$ for all $v_1 v_2 \in E$. Then $f^*$ again induces a labeling say $f^{* *}: V \rightarrow A$ defined by $f^{* *}(v)=\sum_{v v_1 \in E} f^*\left(v v_1\right)$. A graph $G=(V, E)$ is said to be an Induced $A$-Magic Graph (IAMG) if there exists a non zero labeling $f: V \rightarrow A$ such that $f \equiv f^{* *}$. The function $f$, so obtained is called an Induced $A$-Magic Labeling (IAML) of $G$ and a graph which has no such Induced Magic Labeling is called a Non-induced magic graph. In this paper we discuss the existence of Induced Magic Labeling of some special graphs like $P_n, C_n, K_n$ and $K_{m, n}$.

Let $V_4=\{0, a, b, c\}$ be the Klein-4-group with identity element 0 .A graph $G(V(G), E(G))$ is said to be neighbourhood $V_4$-magic if there exists a labeling $f: V(G) \rightarrow V_4 \backslash\{0\}$ such that the induced mapping $N_f^{+}: V(G) \rightarrow V_4$ defined by $N_f^{+}(v)=\sum_{u \in N(v)} f(u)$ is a constant map. If this constant is $p(p \neq 0)$, we say that $f$ is a $p$-neighbourhood $V_4$-magic labeling of $G$ and $G$ a $p$-neighbourhood $V_4$-magic graph. If this constant is zero, we say that $f$ is a 0 -neighghbourhood $V_4$-magic labeling of $G$ and $G$ a 0 -neighbourhood $V_4$-magic graph. In this paper we investigate middle graph of some special graphs that are $a$-neighbourhood $V_4$-magic, 0 -neighbourhood $V_4$-magic and both $a$-neighbourhood and 0 -neighbourhood $V_4$-magic.