On the Second-Order Wiener Ratios of Iterated Line Graphs

The Wiener index W(G) of a graph G is the sum of distances between all unordered pairs of its vertices. Dobrynin and Mel'nikov [in: Distance in Molecular Graphs - Theory, 2012, p. 85-121] propose the study of estimates for extremal values of the ratio R_k(G) = W(L^k(G))/W(G) where L^k(G) denotes the k-th iterated line graph of G. Hri\v{n}\'akov\'a, Knor and \v{S}krekovski [Art Discrete Appl. Math. 1 (2018) #P1.09] prove that for each k>2, the path P_n has the smallest value of the ratio R_k among all trees of large order n, and they conjecture that the same holds for the case k=2. We give a counterexample of every order n>21 to this conjecture

Reinforcement learning for graph theory, II. Small Ramsey numbers

We describe here how the recent Wagner's approach for applying reinforcement learning to construct examples in graph theory can be used in the search for critical graphs for small Ramsey numbers. We illustrate this application by providing lower bounds for the small Ramsey numbers $R(K_{2,5}, K_{3,5})$, $R(B_3, B_6)$ and $R(B_4, B_5)$ and by improving the lower known bound for $R(W_5, W_7)$.Comment: 6 page

Searching for regular, triangle-distinct graphs

The triangle-degree of a vertex v of a simple graph G is the number of triangles in G that contain v. A simple graph is triangle-distinct if all its vertices have distinct triangle-degrees. Berikkyzy et al. [Discrete Math. 347 (2024) 113695] recently asked whether there exists a regular graph that is triangle-distinct. Here we showcase the examples of regular, triangle-distinct graphs with orders between 21 and 27, and report on the methodology used to find them.Comment: 7 pages + 11 pages appendix with examples of regular, triangle-distinct graph

Theorems and computations in circular colourings of graphs

The circular chromatic number provides a more refined measure of colourability of graphs, than does the ordinary chromatic number. Thus circular colouring is of substantial importance wherever graph colouring is studied or applied, for example, to scheduling problems of periodic nature. Precisely, the circular chromatic number of a graph G is the smallest ratio p/q of positive integers p and q for which there exists a mapping c:V(G)->{1,2,...,p} such that q<=|c(u)-c(v)|<=p-q for every edge uv of G. We present some known and new results regarding the computation of the circular chromatic number. In particular, we prove a lemma which can be used to improve the ratio of some circular colourings. These results are later used to bound the circular chromatic number of the plane unit-distance graph, the projective plane orthogonality graph, generalized Petersen graphs, and squares of graphs. Some of the computations in this thesis are computer assisted. Nesetril\u27s "pentagon problem", asks whether the circular chromatic number of every cubic graph having sufficiently high girth is at most 5/2. We prove that the statement of the pentagon colouring problem is false with odd-girth in place of girth; and that if the pentagon colouring problem is true then the girth requirement is at least 10. Additionally, we present results of extensive computations of the circular chromatic numbers of small cubic graphs with girth at most 10. We also prove that every subcubic graph with girth at least 9 which can be embedded in either the plane, the projec tive plane, the torus, or the Klein bottle, has circular chromatic number strictly less than 3. Finally we investigate circular edge colourings of cubic graphs. In particular, we establish the circular chromatic index for several infinite families of snarks, namely Isaacs\u27 flower snarks, Goldberg snarks, and generalized Blanusa snarks

The Circular Chromatic Index of Goldberg

We determine the exact values of the circular chromatic index of the Goldberg snarks, and of a related family, the twisted Goldberg snarks. Key words: Edge colouring, Circular colouring, Snark, Goldberg snarks

Circular Chromatic Index of Generalized Blanusa Snarks

In his Master’s thesis, Ján Mazák proved that the circular chromatic index of the type 1 generalized Blanuˇsa snark B1 n equals 3 + 2 n. This result provided the first infinite set of values of the circular chromatic index of snarks. In this paper we show the type 2 generalized Blanuˇsa snark B 2 n has circular chromatic index 3 + 1 ⌊1+3n/2 ⌋. In particular, this proves that all numbers 3 + 1/n with n � 2 are realized as the circular chromatic index of a snark

A Linear Algebraic Threshold Essential Secret Image Sharing Scheme

A secret sharing scheme allocates to each participant a share of a secret in such a way that authorized subsets of participants can reconstruct the secret, while shares of unauthorized subsets of participants provide no useful information about the secret. For positive integers r,s,t,n with r⩽s⩽t⩽n, an (r,s,t,n)–threshold essential secret sharing scheme is an algorithm that decomposes a secret S into n shares, s of which are essential, in a way that authorized subsets are precisely those with at least t members, at least r of whom are essential. This work proposes a lossless linear algebraic (r,s,t,n)–threshold essential secret image sharing scheme that decomposes the secret, S, into equally-sized shares, each of size 1/t the size of S. For each block, B, of S, the scheme assigns to the n participants distinct signature vectors v1,v2,…,vn in the vector space F2αt, where α is a suitable positive integer, typically between 2 and 5, inclusive. These signature vectors must adhere to some admissibility conditions in order to satisfy the secret sharing threshold properties. The decomposition of B into n shares is obtained by partitioning B into t vectors, then computing the share yj of the jth participant (1≤j≤n), as a linear combination of these parts with coefficients from the signature vj. The presented simulations showcase the effectiveness and robustness of the proposed scheme against standard statistical and security attacks. They further demonstrate its superiority with respect to existing schemes

A Probabilistic Chaotic Image Encryption Scheme

This paper proposes a probabilistic image encryption scheme that improves on existing deterministic schemes by using a chaining mode of chaotic maps in a permutation-masking process. Despite its simplicity, the permutation phase destroys any correlation between adjacent pixel values in a meaningful image. The masking phase, however, modifies the pixel values of the image at hand using pseudorandom numbers with some other initiated random numbers so that any slight change in the plain image spreads throughout the corresponding cipher image. These random numbers ensure the generation of distinct cipher images for the same plain image encryption, even if it is encrypted multiple times with the same key, thereby adding some security features. Simulations show that the proposed scheme is robust to common statistical and security threats. Furthermore, the scheme is shown to be competitive with existing image encryption schemes