We discuss the computational complexity of determining the chromatic number of graphs without long induced paths. We prove NP-completeness of deciding whether a P 8-free graph is 5-colorable and of deciding whether a P 12-free graph is 4-colorable. Moreover, we give a polynomial time algorithm for deciding whether a P 5-free graph is 3-colorable.

Guess-and-determine attack is one of the general attacks on stream ciphers. It is a common cryptanalysis tool for evaluating security of stream ciphers. The effectiveness of this attack is based on the number of unknown bits which will be guessed by the attacker to break the cryptosystem. In this work, we present a relation between the minimum numbers of the guessed bits and uniquely restricted...

A graph G is supereulerian if it has a spanning eulerian subgraph. We prove that every 3-edge-connected graph with the circumference at most 11 has a spanning eulerian subgraph if and only if it is not contractible to the Petersen graph. As applications, we determine collections F1, F2 and F3 of graphs to prove each of the following (i) Every 3-connected {K1,3, Z9}-free graph is hamiltonian if ...

The generalized bull is the graph B i , j ${B}_{i,j}$ obtained by attaching endvertices of two disjoint paths lengths $i,j$ to vertices a triangle. We prove that every 3-connected { K 1 3 X } $\{{K}_{1,3},X\}$ -free graph, where ? 6 2 5 4 $X\in \{{B}_{1,6},{B}_{2,5},{B}_{3,4}\}$ Hamilton-connected. results are sharp and complete characterization forbidden induced bulls implying Hamilton-connect...

We characterize all pairs of connected graphs {X,Y } such that each 3-connected {X,Y }-free graph is pancyclic. In particular, we show that if each of the graphs in such a pair {X,Y } has at least four vertices, then one of them is the claw K1,3, while the other is a subgraph of one of six specified graphs.

Graph bundles generalize the notion of covering graphs and graph products. In [8], authors constructed an algorithm that ÿnds a presentation as a nontrivial Cartesian graph bundle for all graphs that are Cartesian graph bundles over triangle-free simple base. In [21], the unique square property is deÿned and it is shown that any equivalence relation possessing the unique square property determi...

Our first main result shows that a graph product of right cancellative monoids is itself right cancellative. If each of the component monoids satisfies the condition that the intersection of two principal left ideals is either principal or empty, then so does the graph product. Our second main result gives a presentation for the inverse hull of such a graph product. We then specialise to the ca...

In the course of extending Grötzsch’s theorem, we prove that every triangle-free graph without a K5-minor is 3-colorable. It has been recently proved that every triangle-free planar graph admits a homomorphism to the Clebsch graph. We also extend this result to the class of triangle-free graphs without a K5-minor. This is related to some conjectures which generalize the Four-Color Theorem. Whil...

The composition graph of a family of n+1 disjoint graphs {Hi : 0 ≤ i ≤ n} is the graph H obtained by substituting the n vertices of H0 respectively by the graphs H1, H2, ..., Hn. If H has some hereditary property P , then necessarily all its factors enjoy the same property. For some sort of graphs it is sufficient that all factors {Hi : 0 ≤ i ≤ n} have a certain common P to endow H with this P ...

A graph language L is in the class C-edNCE of context-free edNCE graph languages if and only if L = f(T) where f is a function on graphs that can be deened in monadic second-order logic and T is the set of all trees over some ranked alphabet. This logical characterization implies a large number of closure and decidability properties of the context-free edNCE graph languages. Rather than context...