Schrijver graphs are vertex-color-critical subgraphs of Kneser graphs having the same chromatic n... more Schrijver graphs are vertex-color-critical subgraphs of Kneser graphs having the same chromatic number. They also share the value of their fractional chromatic number but Schrijver graphs are not critical for that. Here we present an induced subgraph of every Schrijver graph that is vertex-critical with respect to the fractional chromatic number. These subgraphs turn out to be isomorphic with certain circular complete graphs. We also characterize the critical edges within this subgraph.
Combining Ky Fan's theorem with ideas of Greene and Matoušek we prove a generalization of Dol'nik... more Combining Ky Fan's theorem with ideas of Greene and Matoušek we prove a generalization of Dol'nikov's theorem. Using another variant of the Borsuk-Ulam theorem due to Bacon and Tucker, we also prove the presence of all possible completely multicolored t-vertex complete bipartite graphs in t-colored t-chromatic Kneser graphs and in several of their relatives. In particular, this implies a generalization of a recent result of G. Spencer and F. E. Su.
A coloring is called s-wide if no walk of length 2s -1 connects vertices of the same color. A gra... more A coloring is called s-wide if no walk of length 2s -1 connects vertices of the same color. A graph is s-widely colorable with t colors if and only if it admits a homomorphism into a universal graph W (s, t). Tardif observed that the value of the r th multichromatic number χ r (W (s, t)) of these graphs is at least t + 2(r -1) and equality holds for r = s = 2. He asked whether there is equality also for r = s = 3. We show that χ s (W (s, t)) = t+2(s-1) for all s thereby answering Tardif's question. We observe that for large r (with respect to s and t fixed) we cannot have equality and that for s fixed and t going to infinity the fractional chromatic number of W (s, t) also tends to infinity. The latter is a simple consequence of another result of Tardif on the fractional chromatic number of generalized Mycielski graphs.
An edge-coloring of the complete graph K n we call F -caring if it leaves no F -subgraph of K n m... more An edge-coloring of the complete graph K n we call F -caring if it leaves no F -subgraph of K n monochromatic and at the same time every subset of |V (F )| vertices contains in it at least one completely multicolored version of F . For the first two meaningful cases, when F = K 1,3 and F = P 4 we determine for infinitely many n the minimum number of colors needed for an F -caring edge-coloring of K n . An explicit family of 2⌈log 2 n⌉ 3-edgecolorings of K n so that every quadruple of its vertices contains a totally multicolored P 4 in at least one of them is also presented. Investigating related Ramsey-type problems we also show that the Shannon (OR-)capacity of the Grötzsch graph is strictly larger than that of the five length cycle.
The local chromatic number of a graph, introduced by Erdős et al. in [3], is the minimum number o... more The local chromatic number of a graph, introduced by Erdős et al. in [3], is the minimum number of colors that must appear in the closed neighborhood of some vertex in any proper coloring of the graph. This talk, based on the papers , would like to survey some of our recent results on this parameter. We give a lower bound for the local chromatic number in terms of the lower bound of the chromatic number provided by the topological method introduced by Lovász. We show that this bound is tight in many cases. In particular, we determine the local chromatic number of certain odd chromatic Schrijver graphs and generalized Mycielski graphs. We further elaborate on the case of 4-chromatic graphs and, in particular, on surface quadrangulations.
Schrijver graphs are vertex-color-critical subgraphs of Kneser graphs having the same chromatic n... more Schrijver graphs are vertex-color-critical subgraphs of Kneser graphs having the same chromatic number. They also share the value of their fractional chromatic number but Schrijver graphs are not critical for that. Here we present an induced subgraph of every Schrijver graph that is vertex-critical with respect to the fractional chromatic number. These subgraphs turn out to be isomorphic with certain circular complete graphs. We also characterize the critical edges within this subgraph.
Let D ⊆ N be an arbitrary subset of the natural numbers. For every n, let M (n, D) be the maximum... more Let D ⊆ N be an arbitrary subset of the natural numbers. For every n, let M (n, D) be the maximum of the cardinality of a set of Hamiltonian paths in the complete graph K n such that the union of any two paths from the family contains a not necessarily induced cycle of some length from D. We determine or bound the asymptotics of M (n, D) in various special cases. This problem is closely related to that of the permutation capacity of graphs and constitutes a further extension of the problem area around Shannon capacity. We also discuss how to generalize our cycle-difference problems and present an example where cycles are replaced by 4-cliques. These problems are in a natural duality to those of graph intersection, initiated by Erdős, Simonovits and Sós. The lack of kernel structure as a natural candidate for optimum makes our problems quite challenging. * A talk based on the material in this paper is accepted for presentation at the EuroComb'11 conference.
The local chromatic number of a graph G is the number of colors appearing in the most colorful cl... more The local chromatic number of a graph G is the number of colors appearing in the most colorful closed neighborhood of a vertex minimized over all proper colorings of G. We show that two specific topological obstructions that have the same implications for the chromatic number have different implications for the local chromatic number. These two obstructions can be formulated in terms of the homomorphism complex Hom(K 2 , G) and its suspension, respectively. These investigations follow the line of research initiated by Matoušek and Ziegler who recognized a hierarchy of the different topological expressions that can serve as lower bounds for the chromatic number of a graph. Our results imply that the local chromatic number of 4-chromatic Kneser, Schrijver, Borsuk, and generalized Mycielski graphs is 4, and more generally, that 2r-chromatic versions of these graphs have local chromatic number at least r + 2. This lower bound is tight in several cases by results in .
Combining Ky Fan's theorem with ideas of Greene and Matoušek we prove a generalization of Dol'nik... more Combining Ky Fan's theorem with ideas of Greene and Matoušek we prove a generalization of Dol'nikov's theorem. Using another variant of the Borsuk-Ulam theorem due to Bacon and Tucker, we also prove the presence of all possible completely multicolored t-vertex complete bipartite graphs in t-colored t-chromatic Kneser graphs and in several of their relatives. In particular, this implies a generalization of a recent result of G. Spencer and F. E. Su.
Bibliography 115 ii Bevezetésésáttekintés (Summary in Hungarian) Az egyik legfontosabbés legtöbbe... more Bibliography 115 ii Bevezetésésáttekintés (Summary in Hungarian) Az egyik legfontosabbés legtöbbet vizsgált gráfelméleti paraméter a kromatikus szám, azon színek minimális száma, melyekkel a gráf csúcsai kiszínezhet˝ okúgy, hogy szomszédos csúcsok színe különböz˝ o legyen. A G gráf χ(G) kromatikus számának egyik legtermészetesebb alsó becslése a gráf ω(G) klikkszáma, a csúcshalmaz legnagyobb olyan részhalmazának elemszáma, amelyben min-den csúcspárössze van kötve. Ez a becslés sokszor nagyon gyenge, egy háromszögmentes gráf kromatikus száma is lehet tetsz˝ olegesen nagy, ezt bizonyítja pl. Mycielski [123] konstrukciója. Azértekezésben fontos szerepet játszik gráfok konormálisnak nevezett szorzata, illetve hatványa, melynek bevezetését egyebek mellett számos információelméleti kérdés mo-tiválja. E m˝ uvelet során a klikkszámés a kromatikus szám ellentétesen viselkedik: el˝ obbi szupermultiplikatív, utóbbi szubmultiplikatív erre a szorzásra nézve, melynek definíciója a következ˝ o. G ...
Discrete Mathematics & Theoretical Computer Science, 2005
The local chromatic number of a graph, introduced by Erdős et al., is the minimum number of color... more The local chromatic number of a graph, introduced by Erdős et al., is the minimum number of colors that must appear in the closed neighborhood of some vertex in any proper coloring of the graph. This talk would like to survey some of our recent results on this parameter. We give a lower bound for the local chromatic number in terms of the lower bound of the chromatic number provided by the topological method introduced by Lovász. We show that this bound is tight in many cases. In particular, we determine the local chromatic number of certain odd chromatic Schrijver graphs and generalized Mycielski graphs. We further elaborate on the case of $4$-chromatic graphs and, in particular, on surface quadrangulations.
Perfect hash functions, superimposed codes as well as some other fashionable questions in compute... more Perfect hash functions, superimposed codes as well as some other fashionable questions in computer science and random-access communication are special cases of early-day information theoretic models in the zero-error case. A new class of problems in asymptotic combinatorics can be formulated as the determination of the zero-error capacity of a class of discrete memoryless channels. (This model is also known as the compound channel). We solve an interesting class of these problems using our recent results in polyhedral combinatorics. R. M. Capocelli (ed.
Combining Ky Fan's theorem with ideas of Greene and Matoušek we prove a generalization of Dol'nik... more Combining Ky Fan's theorem with ideas of Greene and Matoušek we prove a generalization of Dol'nikov's theorem. Using another variant of the Borsuk-Ulam theorem due to Tucker and Bacon, we also prove the presence of all possible completely multicolored t-vertex complete bipartite graphs in t-colored t-chromatic Kneser graphs and in several of their relatives. In particular, this implies a generalization of a recent result of G. Spencer and F. E. Su.
We give an elementary proof for the non-3-colorability of 4-chromatic Schrijver graphs thus provi... more We give an elementary proof for the non-3-colorability of 4-chromatic Schrijver graphs thus providing such a proof also for 4-chromatic Kneser graphs. To this end we use a complete description of the structure of 4-chromatic Schrijver graphs that was already given by Braun in and even earlier in an unpublished manuscript by Li . We also address connections to surface quadrangulations. In particular, we show that a spanning subgraph of 4-chromatic Schrijver graphs quadrangulates the Klein bottle, while another spanning subgraph quadrangulates the projective plane. The latter is a special case of a result by Kaiser and Stehlík [11]. We characterize the color-critical edges of 4-chromatic Schrijver graphs and also present preliminary results toward the characterization of color-critical edges in Schrijver graphs of higher chromatic number. Finally, we show that (apart from two cases of small parameters) the subgraphs we present that quadrangulate the Klein bottle are edge-color-critical. The analogous result for the subgraphs quadrangulating the projective plane is an immediate consequence of earlier results by Gimbel and Thomassen and was already noted by Kaiser and Stehlík.
The local chromatic number of a graph was introduced in [4]. In [17] a connection to topological ... more The local chromatic number of a graph was introduced in [4]. In [17] a connection to topological properties of (a box complex of) the graph was established and in [18] it was shown that if a graph is strongly topologically 4-chromatic then its local chromatic number is at least four. As a consequence one obtains a generalization of the following theorem of Youngs [19]: If a quadrangulation of the projective plane is not bipartite it has chromatic number four. The generalization states that in this case the local chromatic number is also four. Both papers [1] and [13] generalize Youngs's result to arbitrary non-orientable surfaces replacing the condition of the graph being not bipartite by a more technical condition of an odd quadrangulation. This paper investigates when these general results are true for the local chromatic number instead of the chromatic number. Surprisingly, we find out that (unlike in the case of the chromatic number) this depends on the genus of the surface. For the non-orientable surfaces of genus at most four, the local chromatic number of any odd quadrangulation is at least four, but this is not true for non-orientable surfaces of genus 5 or higher. We also prove that face subdivisions of odd quadrangulations and Fisk triangulations of arbitrary surfaces exhibit the same behavior for the local chromatic number as they do for the usual chromatic number.
A coloring is called s-wide if no walk of length 2s − 1 connects vertices of the same color. A gr... more A coloring is called s-wide if no walk of length 2s − 1 connects vertices of the same color. A graph is s-widely colorable with t colors if and only if it admits a homomorphism into a universal graph W (s, t). Tardif observed that the value of the r th multichromatic number χ r (W (s, t)) of these graphs is at least t + 2(r − 1) and equality holds for r = s = 2. He asked whether there is equality also for r = s = 3. We show that χ s (W (s, t)) = t+2(s−1) for all s thereby answering Tardif's question. We observe that for large r (with respect to s and t fixed) we cannot have equality and that for s fixed and t going to infinity the fractional chromatic number of W (s, t) also tends to infinity. The latter is a simple consequence of another result of Tardif on the fractional chromatic number of generalized Mycielski graphs.
We investigate the maximum size of graph families on a common vertex set of cardinality n such th... more We investigate the maximum size of graph families on a common vertex set of cardinality n such that the symmetric difference of the edge sets of any two members of the family satisfies some prescribed condition. We solve the problem completely for infinitely many values of n when the prescribed condition is connectivity or 2-connectivity, Hamiltonicity or the containment of a spanning star. We also investigate local conditions that can be certified by looking at only a subset of the vertex set. In these cases a capacity-type asymptotic invariant is defined and when the condition is to contain a certain subgraph this invariant is shown to be a simple function of the chromatic number of this required subgraph. This is proven using classical results from extremal graph theory. Several variants are considered and the paper ends with a collection of open problems.
We investigate a communication setup where a source output is sent through a free noisy channel f... more We investigate a communication setup where a source output is sent through a free noisy channel first and an additional codeword is sent through a noiseless but expensive channel later. With the help of the second message the decoder should be able to decide with zero-error whether its decoding of the first message was error-free. This scenario leads to the definition of a digraph parameter that generalizes Witsenhausen's zero-error rate for directed graphs. We investigate this new parameter for some specific directed graphs and explore its relations to other digraph parameters like Sperner capacity and dichromatic number. When the original problem is modified to require zero-error decoding of the whole message then we arrive back to the Witsenhausen rate of an appropriately defined undirected graph.
There are several famous unsolved conjectures about the chromatic number that were relaxed and al... more There are several famous unsolved conjectures about the chromatic number that were relaxed and already proven to hold for the fractional chromatic number. We discuss similar relaxations for the topological lower bound(s) of the chromatic number. In particular, we prove that such a relaxed version is true for the Behzad-Vizing conjecture and also discuss the conjectures of Hedetniemi and of Hadwiger from this point of view. For the latter, a similar statement was already proven in an earlier paper of the first author with G. Tardos, our main concern here is that the so-called odd Hadwiger conjecture looks much more difficult in this respect. We prove that the statement of the odd Hadwiger conjecture holds for large enough Kneser graphs and Schrijver graphs of any fixed chromatic number.
We show that the minimum number of orientations of the edges of the n-vertex complete graph havin... more We show that the minimum number of orientations of the edges of the n-vertex complete graph having the property that every triangle is made cyclic in at least one of them is _2(n-1). More generally, we also determine the minimum number of orientations of K_n such that at least one of them orients some specific k-cycles cyclically on every k-element subset of the vertex set. The questions answered by these results were motivated by an analogous problem of Vera T. Sós concerning triangles and 3-edge-colorings. Some variants of the problem are also considered.
Schrijver graphs are vertex-color-critical subgraphs of Kneser graphs having the same chromatic n... more Schrijver graphs are vertex-color-critical subgraphs of Kneser graphs having the same chromatic number. They also share the value of their fractional chromatic number but Schrijver graphs are not critical for that. Here we present an induced subgraph of every Schrijver graph that is vertex-critical with respect to the fractional chromatic number. These subgraphs turn out to be isomorphic with certain circular complete graphs. We also characterize the critical edges within this subgraph.
Combining Ky Fan's theorem with ideas of Greene and Matoušek we prove a generalization of Dol'nik... more Combining Ky Fan's theorem with ideas of Greene and Matoušek we prove a generalization of Dol'nikov's theorem. Using another variant of the Borsuk-Ulam theorem due to Bacon and Tucker, we also prove the presence of all possible completely multicolored t-vertex complete bipartite graphs in t-colored t-chromatic Kneser graphs and in several of their relatives. In particular, this implies a generalization of a recent result of G. Spencer and F. E. Su.
A coloring is called s-wide if no walk of length 2s -1 connects vertices of the same color. A gra... more A coloring is called s-wide if no walk of length 2s -1 connects vertices of the same color. A graph is s-widely colorable with t colors if and only if it admits a homomorphism into a universal graph W (s, t). Tardif observed that the value of the r th multichromatic number χ r (W (s, t)) of these graphs is at least t + 2(r -1) and equality holds for r = s = 2. He asked whether there is equality also for r = s = 3. We show that χ s (W (s, t)) = t+2(s-1) for all s thereby answering Tardif's question. We observe that for large r (with respect to s and t fixed) we cannot have equality and that for s fixed and t going to infinity the fractional chromatic number of W (s, t) also tends to infinity. The latter is a simple consequence of another result of Tardif on the fractional chromatic number of generalized Mycielski graphs.
An edge-coloring of the complete graph K n we call F -caring if it leaves no F -subgraph of K n m... more An edge-coloring of the complete graph K n we call F -caring if it leaves no F -subgraph of K n monochromatic and at the same time every subset of |V (F )| vertices contains in it at least one completely multicolored version of F . For the first two meaningful cases, when F = K 1,3 and F = P 4 we determine for infinitely many n the minimum number of colors needed for an F -caring edge-coloring of K n . An explicit family of 2⌈log 2 n⌉ 3-edgecolorings of K n so that every quadruple of its vertices contains a totally multicolored P 4 in at least one of them is also presented. Investigating related Ramsey-type problems we also show that the Shannon (OR-)capacity of the Grötzsch graph is strictly larger than that of the five length cycle.
The local chromatic number of a graph, introduced by Erdős et al. in [3], is the minimum number o... more The local chromatic number of a graph, introduced by Erdős et al. in [3], is the minimum number of colors that must appear in the closed neighborhood of some vertex in any proper coloring of the graph. This talk, based on the papers , would like to survey some of our recent results on this parameter. We give a lower bound for the local chromatic number in terms of the lower bound of the chromatic number provided by the topological method introduced by Lovász. We show that this bound is tight in many cases. In particular, we determine the local chromatic number of certain odd chromatic Schrijver graphs and generalized Mycielski graphs. We further elaborate on the case of 4-chromatic graphs and, in particular, on surface quadrangulations.
Schrijver graphs are vertex-color-critical subgraphs of Kneser graphs having the same chromatic n... more Schrijver graphs are vertex-color-critical subgraphs of Kneser graphs having the same chromatic number. They also share the value of their fractional chromatic number but Schrijver graphs are not critical for that. Here we present an induced subgraph of every Schrijver graph that is vertex-critical with respect to the fractional chromatic number. These subgraphs turn out to be isomorphic with certain circular complete graphs. We also characterize the critical edges within this subgraph.
Let D ⊆ N be an arbitrary subset of the natural numbers. For every n, let M (n, D) be the maximum... more Let D ⊆ N be an arbitrary subset of the natural numbers. For every n, let M (n, D) be the maximum of the cardinality of a set of Hamiltonian paths in the complete graph K n such that the union of any two paths from the family contains a not necessarily induced cycle of some length from D. We determine or bound the asymptotics of M (n, D) in various special cases. This problem is closely related to that of the permutation capacity of graphs and constitutes a further extension of the problem area around Shannon capacity. We also discuss how to generalize our cycle-difference problems and present an example where cycles are replaced by 4-cliques. These problems are in a natural duality to those of graph intersection, initiated by Erdős, Simonovits and Sós. The lack of kernel structure as a natural candidate for optimum makes our problems quite challenging. * A talk based on the material in this paper is accepted for presentation at the EuroComb'11 conference.
The local chromatic number of a graph G is the number of colors appearing in the most colorful cl... more The local chromatic number of a graph G is the number of colors appearing in the most colorful closed neighborhood of a vertex minimized over all proper colorings of G. We show that two specific topological obstructions that have the same implications for the chromatic number have different implications for the local chromatic number. These two obstructions can be formulated in terms of the homomorphism complex Hom(K 2 , G) and its suspension, respectively. These investigations follow the line of research initiated by Matoušek and Ziegler who recognized a hierarchy of the different topological expressions that can serve as lower bounds for the chromatic number of a graph. Our results imply that the local chromatic number of 4-chromatic Kneser, Schrijver, Borsuk, and generalized Mycielski graphs is 4, and more generally, that 2r-chromatic versions of these graphs have local chromatic number at least r + 2. This lower bound is tight in several cases by results in .
Combining Ky Fan's theorem with ideas of Greene and Matoušek we prove a generalization of Dol'nik... more Combining Ky Fan's theorem with ideas of Greene and Matoušek we prove a generalization of Dol'nikov's theorem. Using another variant of the Borsuk-Ulam theorem due to Bacon and Tucker, we also prove the presence of all possible completely multicolored t-vertex complete bipartite graphs in t-colored t-chromatic Kneser graphs and in several of their relatives. In particular, this implies a generalization of a recent result of G. Spencer and F. E. Su.
Bibliography 115 ii Bevezetésésáttekintés (Summary in Hungarian) Az egyik legfontosabbés legtöbbe... more Bibliography 115 ii Bevezetésésáttekintés (Summary in Hungarian) Az egyik legfontosabbés legtöbbet vizsgált gráfelméleti paraméter a kromatikus szám, azon színek minimális száma, melyekkel a gráf csúcsai kiszínezhet˝ okúgy, hogy szomszédos csúcsok színe különböz˝ o legyen. A G gráf χ(G) kromatikus számának egyik legtermészetesebb alsó becslése a gráf ω(G) klikkszáma, a csúcshalmaz legnagyobb olyan részhalmazának elemszáma, amelyben min-den csúcspárössze van kötve. Ez a becslés sokszor nagyon gyenge, egy háromszögmentes gráf kromatikus száma is lehet tetsz˝ olegesen nagy, ezt bizonyítja pl. Mycielski [123] konstrukciója. Azértekezésben fontos szerepet játszik gráfok konormálisnak nevezett szorzata, illetve hatványa, melynek bevezetését egyebek mellett számos információelméleti kérdés mo-tiválja. E m˝ uvelet során a klikkszámés a kromatikus szám ellentétesen viselkedik: el˝ obbi szupermultiplikatív, utóbbi szubmultiplikatív erre a szorzásra nézve, melynek definíciója a következ˝ o. G ...
Discrete Mathematics & Theoretical Computer Science, 2005
The local chromatic number of a graph, introduced by Erdős et al., is the minimum number of color... more The local chromatic number of a graph, introduced by Erdős et al., is the minimum number of colors that must appear in the closed neighborhood of some vertex in any proper coloring of the graph. This talk would like to survey some of our recent results on this parameter. We give a lower bound for the local chromatic number in terms of the lower bound of the chromatic number provided by the topological method introduced by Lovász. We show that this bound is tight in many cases. In particular, we determine the local chromatic number of certain odd chromatic Schrijver graphs and generalized Mycielski graphs. We further elaborate on the case of $4$-chromatic graphs and, in particular, on surface quadrangulations.
Perfect hash functions, superimposed codes as well as some other fashionable questions in compute... more Perfect hash functions, superimposed codes as well as some other fashionable questions in computer science and random-access communication are special cases of early-day information theoretic models in the zero-error case. A new class of problems in asymptotic combinatorics can be formulated as the determination of the zero-error capacity of a class of discrete memoryless channels. (This model is also known as the compound channel). We solve an interesting class of these problems using our recent results in polyhedral combinatorics. R. M. Capocelli (ed.
Combining Ky Fan's theorem with ideas of Greene and Matoušek we prove a generalization of Dol'nik... more Combining Ky Fan's theorem with ideas of Greene and Matoušek we prove a generalization of Dol'nikov's theorem. Using another variant of the Borsuk-Ulam theorem due to Tucker and Bacon, we also prove the presence of all possible completely multicolored t-vertex complete bipartite graphs in t-colored t-chromatic Kneser graphs and in several of their relatives. In particular, this implies a generalization of a recent result of G. Spencer and F. E. Su.
We give an elementary proof for the non-3-colorability of 4-chromatic Schrijver graphs thus provi... more We give an elementary proof for the non-3-colorability of 4-chromatic Schrijver graphs thus providing such a proof also for 4-chromatic Kneser graphs. To this end we use a complete description of the structure of 4-chromatic Schrijver graphs that was already given by Braun in and even earlier in an unpublished manuscript by Li . We also address connections to surface quadrangulations. In particular, we show that a spanning subgraph of 4-chromatic Schrijver graphs quadrangulates the Klein bottle, while another spanning subgraph quadrangulates the projective plane. The latter is a special case of a result by Kaiser and Stehlík [11]. We characterize the color-critical edges of 4-chromatic Schrijver graphs and also present preliminary results toward the characterization of color-critical edges in Schrijver graphs of higher chromatic number. Finally, we show that (apart from two cases of small parameters) the subgraphs we present that quadrangulate the Klein bottle are edge-color-critical. The analogous result for the subgraphs quadrangulating the projective plane is an immediate consequence of earlier results by Gimbel and Thomassen and was already noted by Kaiser and Stehlík.
The local chromatic number of a graph was introduced in [4]. In [17] a connection to topological ... more The local chromatic number of a graph was introduced in [4]. In [17] a connection to topological properties of (a box complex of) the graph was established and in [18] it was shown that if a graph is strongly topologically 4-chromatic then its local chromatic number is at least four. As a consequence one obtains a generalization of the following theorem of Youngs [19]: If a quadrangulation of the projective plane is not bipartite it has chromatic number four. The generalization states that in this case the local chromatic number is also four. Both papers [1] and [13] generalize Youngs's result to arbitrary non-orientable surfaces replacing the condition of the graph being not bipartite by a more technical condition of an odd quadrangulation. This paper investigates when these general results are true for the local chromatic number instead of the chromatic number. Surprisingly, we find out that (unlike in the case of the chromatic number) this depends on the genus of the surface. For the non-orientable surfaces of genus at most four, the local chromatic number of any odd quadrangulation is at least four, but this is not true for non-orientable surfaces of genus 5 or higher. We also prove that face subdivisions of odd quadrangulations and Fisk triangulations of arbitrary surfaces exhibit the same behavior for the local chromatic number as they do for the usual chromatic number.
A coloring is called s-wide if no walk of length 2s − 1 connects vertices of the same color. A gr... more A coloring is called s-wide if no walk of length 2s − 1 connects vertices of the same color. A graph is s-widely colorable with t colors if and only if it admits a homomorphism into a universal graph W (s, t). Tardif observed that the value of the r th multichromatic number χ r (W (s, t)) of these graphs is at least t + 2(r − 1) and equality holds for r = s = 2. He asked whether there is equality also for r = s = 3. We show that χ s (W (s, t)) = t+2(s−1) for all s thereby answering Tardif's question. We observe that for large r (with respect to s and t fixed) we cannot have equality and that for s fixed and t going to infinity the fractional chromatic number of W (s, t) also tends to infinity. The latter is a simple consequence of another result of Tardif on the fractional chromatic number of generalized Mycielski graphs.
We investigate the maximum size of graph families on a common vertex set of cardinality n such th... more We investigate the maximum size of graph families on a common vertex set of cardinality n such that the symmetric difference of the edge sets of any two members of the family satisfies some prescribed condition. We solve the problem completely for infinitely many values of n when the prescribed condition is connectivity or 2-connectivity, Hamiltonicity or the containment of a spanning star. We also investigate local conditions that can be certified by looking at only a subset of the vertex set. In these cases a capacity-type asymptotic invariant is defined and when the condition is to contain a certain subgraph this invariant is shown to be a simple function of the chromatic number of this required subgraph. This is proven using classical results from extremal graph theory. Several variants are considered and the paper ends with a collection of open problems.
We investigate a communication setup where a source output is sent through a free noisy channel f... more We investigate a communication setup where a source output is sent through a free noisy channel first and an additional codeword is sent through a noiseless but expensive channel later. With the help of the second message the decoder should be able to decide with zero-error whether its decoding of the first message was error-free. This scenario leads to the definition of a digraph parameter that generalizes Witsenhausen's zero-error rate for directed graphs. We investigate this new parameter for some specific directed graphs and explore its relations to other digraph parameters like Sperner capacity and dichromatic number. When the original problem is modified to require zero-error decoding of the whole message then we arrive back to the Witsenhausen rate of an appropriately defined undirected graph.
There are several famous unsolved conjectures about the chromatic number that were relaxed and al... more There are several famous unsolved conjectures about the chromatic number that were relaxed and already proven to hold for the fractional chromatic number. We discuss similar relaxations for the topological lower bound(s) of the chromatic number. In particular, we prove that such a relaxed version is true for the Behzad-Vizing conjecture and also discuss the conjectures of Hedetniemi and of Hadwiger from this point of view. For the latter, a similar statement was already proven in an earlier paper of the first author with G. Tardos, our main concern here is that the so-called odd Hadwiger conjecture looks much more difficult in this respect. We prove that the statement of the odd Hadwiger conjecture holds for large enough Kneser graphs and Schrijver graphs of any fixed chromatic number.
We show that the minimum number of orientations of the edges of the n-vertex complete graph havin... more We show that the minimum number of orientations of the edges of the n-vertex complete graph having the property that every triangle is made cyclic in at least one of them is _2(n-1). More generally, we also determine the minimum number of orientations of K_n such that at least one of them orients some specific k-cycles cyclically on every k-element subset of the vertex set. The questions answered by these results were motivated by an analogous problem of Vera T. Sós concerning triangles and 3-edge-colorings. Some variants of the problem are also considered.
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Papers by Gábor Simonyi